MVR4 I/questionset3_1:6#%"nG62%"AG%#A1G%#A2G%#A3G%#A4G%"bG%"BG%"vG%#v1G%#v2G%#v3G%#v 4G%"wG%#eqG%$eq1G%$eq2G6"F7C%@$/%%flagG""!C$-%'printfG6$%4QUESTION~NUMBER~%d|+G %&countG-%&blankG6#"""-F?6#%?|+****************************|+G@A/9$FFC(-F?6#%jn The~set~of~all~2x2~matrices~with~respect~to~the~usual~addition|+G-F?6#%Gand~sca lar~multiplication~of~matrices|+G-F?6#%:(a)~forms~a~linear~space|+G-F?6#%B(b)~d oes~not~form~a~linear~space|+G-F?6#%B(c)~is~not~closed~under~addition|+G-F?6#%O (d)~is~not~closed~under~scalar~multiplication|+G/FL""#C*-F?6#%IThe~set~of~all~2 x2~matrices~of~the~form|+G-%&printG6#/8$-%&evalmG6#-%'matrixG6#7$7$%"aGFF7$%"cG %"dG-F?6#%\owhere~a,~b,~and~c~are~arbitrary~real~numbers,with~respect~to~the|+G -F?6#%Vusual~addition~and~scalar~multiplication~of~matrices|+GFTFW-F?6#%>(c)~is ~closed~under~addition|+G-F?6#%K(d)~is~closed~under~scalar~multiplication|+G/FL ""$C*F]o-Fao6#/Fdo-Ffo6#-Fio6#7$7$F]pFFC-F?6#%H|+********************************* ****|+G@AFKC/-F?6#%4Given~the~matrix~A|+G-Fao6#/Fdo-Ffo6#-Fio6#7$7$F[oF`r7$FbsF dt-F?6#%2and~the~matrices|+G>8%-Fio6#7$7$FFF<7$F8&-Fio6#7$7$F8'-Fio 6#7$F`elFfel-Fao6#/F[el-Ffo6#F[el-Fao6#/Fbel-Ffo6#Fbel-Fao6#/Fhel-Ffo6#Fhel-F?6 #%in(a)~The~matrix~A~is~not~a~linear~combination~of~A1,~A2,and~A3|+G-F?6#%fn(b) ~The~matrix~A~is~a~linear~combination~of~A1,~A2,~and~A3|+G-F?6#%S(c)~The~matrix ~A~is~in~the~span~of~A1,~A2,~and~A3|+GFcxFjnC1FjclF]dlFgdl>F[elF\el>FbelFcel>Fh elFiel>8(-Fio6#7$F`elF_elF\flFaflFffl-Fao6#/Figl-Ffo6#Figl-F?6#%\o(a)~The~matri x~A~is~not~a~linear~combination~of~A1,~A2,A3~and~A4|+G-F?6#%hn(b)~The~matrix~A~ is~a~linear~combination~of~A1,~A2,A3~and~A4|+G-F?6#%Z(c)~The~matrix~A~is~not~in ~the~span~of~A1,~A2,~A3~and~A4|+GFcxF]qC)-F?6#%IThe~second~row~of~the~matrix~A~ given~by|+G>Fdo-Fio6#7%7%FFF[oFgv7%F^qFFF`r7%FbsFdt!"(-Fao6#/Fdo-Ffo6#Fdo-F?6#% Q(a)~is~linear~combination~of~the~other~two~rows|+G-F?6#%F(b)~is~a~multiple~of~ the~row3-2*row1|+G-F?6#%W(c)~is~not~a~linear~combination~of~the~other~two~rows| +GFcxF_rC)-F?6#%HThe~third~row~of~the~matrix~A~given~by|+G>Fdo-Fio6#7%Fcil7%F`u Few!"*FeilFgil-F?6#%D(a)~is~a~multiple~of~the~first~row|+G-F?6#%S(b)~is~a~linea r~combination~of~the~other~two~rows|+GFbjlFcxFasC)-F?6#%\oThe~determinant~of~th e~matrix~A~below~is~equal~to~zero.~Then~the|+G>FdoFjjlFgil-F?6#%T(a)~first~row~ is~not~in~the~span~of~the~second~row|+G-F?6#%T(b)~third~row~is~not~in~the~span~ of~the~second~row|+G-F?6#%^o(c)~rows~of~the~matrix~are~not~a~linear~combination ~of~one~another|+G-F?6#%hn(d)~rows~of~the~matrix~are~linear~combination~of~one~ another|+GFctC6-F?6#%,The~vector|+G>80-Fcy6#7%F_]lF^qF`r-Fao6#/F[]m-Ffo6#F[]m-F ?6#%L(a)~is~a~linear~combination~of~the~vectors|+G>F_y-Fcy6#7%FFFgvF[o>Fiy-Fcy6 #7%F]wF`rF[o-Fao6#/F_y-Ffo6#F_y-Fao6#/Fiy-Ffo6#Fiy-F?6#%L(b)~is~a~linear~combin ation~of~the~vectors|+G>F_y-Fcy6#7%FFFFF<>Fiy-Fcy6#7%FF_y-Fcy6#7%FFFgvF<>Fiy-Fcy6#7%FFF F_y-Fcy6#7%FFF Fiy-Fcy6#7%FF_y-Fcy6#7%FFF[oF`r>Fiy-F cy6#7%F]p8)F_pF_^mFd^m-F?6#%Z(a)~v2~is~a~linear~combination~of~v1~if~a=1,b=4,~a nd~c=4|+G-F?6#%Z(b)~v2~is~a~linear~combination~of~v1~if~a=2,b=4,~and~c=8|+G-F?6 #%Z(c)~v2~is~a~linear~combination~of~v1~if~a=1,b=2,~and~c=8|+G-F?6#%Z(d)~v2~is~ a~linear~combination~of~v1~if~a=2,b=2,~and~c=4|+GF[vC)-F?6#%\oWhich~of~the~foll owing~statements~is~true~if~a~vector~w~in~Rn~is|+G-F?6#%Ynot~a~linear~combinati on~of~vectors~v1,~v2,..,vn~in~Rn?|+G-F?6#%fnThe~nonhomogeneous~system~associate d~with~this~combination|+G-F?6#%R(a)~is~consistent~with~infinitely~many~solutio ns|+G-F?6#%5(b)~is~inconsistent|+G-F?6#%Y(c)~the~coefficient~matrix~of~the~syst em~is~nonsingular|+GFcxF^wC)-F?6#%`oWhich~of~the~following~statements~is~true~f or~a~vector~w~in~Rn~to~be|+G-F?6#%`oexpressed~uniquely~as~a~linear~combination~ of~vectors~v1,.,vn~in~Rn?|+GF\cmF_cmFbcm-F?6#%V(c)~the~coefficient~matrix~of~th e~system~is~singular|+G-F?6#%K(d)~the~coefficient~matrix~is~nonsingular|+GFdwC) Ficm-F?6#%doexpressed~in~many~ways~as~a~linear~combination~of~vectors~v1,.,vn~i n~Rn?|+GF\cmF_cmFbcmFecm-F?6#%B(d)~(a)~and~(c)~are~both~correct|+GFfxC/-F?6#%1I f~the~vector~v|+G>Fi\l-Fcy6#7$FgvFbs-Fao6#/Fi\l-Ffo6#Fi\l-F?6#%Tis~in~the~span~ of~the~space~spanned~by~the~vectors|+G>F_y-Fcy6#7$FFFgv>Fiy-Fcy6#7$F[oFFi\l-Fcy6#7%FgvF[oFF_y-Fcy6#7%FFFFiyFe`m>8.Fi`mF_^m Fd^m-Fao6#/F^hm-Ffo6#F^hm-F?6#%GThat~is,~v~=~a*v1~+~b*v2~+~c*v3,~then|+G-F?6#%> (a)~a~=~-1,~~~b~=~~0,~~c~=-2|+G-F?6#%=(b)~a~=~~0,~~~b~=~1,~~c~=~2|+G-F?6#%>(c)~ a~=~-2,~~~b~=~~2,~~c~=-2|+G-F?6#%=(d)~a~=~~0,~~~b~=~-1,~~c~=2|+GFa_lC1-F?6#%.Th e~vector~v|+G-Fao6#/Fi\l-Ffo6#-Fcy6#7%FFF[oF^q-F?6#%3is~in~the~span~of|+G-F?6#% 1(a)~the~vectors|+G-Fao6#/F_y-Ffo6#-Fcy6#7%FgvF^qF`r-Fao6#/Fiy-Ffo6#-Fcy6#7%F[o FgvF<-F?6#%1(b)~the~vectors|+G-Fao6#/F_y-Ffo6#-Fcy6#7%FgvFFC-F?6#%@|+*****************************|+G@ EFKC'-F?6#%BThe~set~|fr~(x1,x2,x3)|gr~x1+x2=x3|hr~|+G-F?6#%Y(a)~is~an~example~o f~a~subspace~spanned~by~one~vector~1|+G-F?6#%Y(b)~is~an~example~of~a~subspace~~ spanned~by~two~vectors|+G-F?6#%H(c)~is~an~example~of~a~subspace~of~R2~|+G-F?6#% hn(d)~is~an~example~of~a~subspace~containing~the~point~(1,1,3)|+GFjnC'-F?6#%hnT he~sets~S1=|fr(x1,x2)|grx1+x2=0|hr~and~S2=|fr(x1,x2)|grx1-x2=0|hr.~Then|+G-F?6# %co(a)~The~intersection~of~S1~and~S2~is~a~subspace~spanned~by~one~vector~1|+G-F ?6#%in(b)~The~intersection~of~S1~and~S2~is~a~subspace~of~dimension~0G-F?6#%[o(c )~The~union~of~S1~and~S2~is~a~subspace~spanned~by~one~vector|+~G-F?6#%[o(d)~The ~union~of~S1~and~S2~is~a~subspace~spanned~by~two~vectors|+GF]qC)-F?6#%FThe~set~ S=|fr~(x1,x2,x3)|gr~x1+x2-x3=1|hr~|+G-F?6#%M(a)~S~is~a~subspace~spanned~by~two~ vectors~|+G-F?6#%gn(b)~S~is~not~a~subspace~because~it~is~closed~under~addition| +G-F?6#%P~~~~but~not~closed~under~scalar~multiplication|+G-F?6#%[o(c)~S~is~not~ a~subspace~because~it~is~not~closed~under~addition|+G-F?6#%L~~~~but~closed~unde r~scalar~multiplication|+G-F?6#%hn(d)~S~is~not~a~subspace~because~the~zero~elem ent~is~not~in~S|+GF_rC'-F?6#%PIf~A~is~a~5x6~matrix,~then~the~row~space~of~A~|+G -F?6#%K(a)~is~a~subspace~spanned~by~six~vectors~|+G-F?6#%K(b)~is~a~subspace~spa nned~by~five~vectors|+G-F?6#%9(c)~is~a~subspace~of~R5|+G-F?6#%9(d)~is~a~subspac e~of~R6|+GFasC'-F?6#%EThe~set~S=|fr(x1,x2,x3)|gr~x1+x2-x3=c|hr~|+G-F?6#%?(a)~S~ is~a~subspace~if~c~=~1~|+G-F?6#%?(b)~S~is~a~subspace~if~c~=~0~|+G-F?6#%O(c)~S~i s~never~a~subspace~for~any~choice~of~c|+G-F?6#%T(d)~is~a~subspace~for~infinitel y~many~choices~of~c|+GFctC(-F?6#%YIf~for~a~given~nxn~system~Ax=b~of~linear~equa tions,~the|+G-F?6#%Pnull~space~of~A~is~trivial.~Then~the~system~is|+G-F?6#%O(a) ~consistent~with~infinitely~many~solutions|+G-F?6#%2(b)~inconsistent|+G-F?6#%G( c)~consistent~with~a~unique~solution|+G-F?6#%M(d)~consistent~only~if~b~is~the~z ero~vector|+GF_uC(-F?6#%YIf~for~a~given~mxn~system~Ax=b~of~linear~equations,~th e|+G-F?6#%Snull~space~of~A~is~nontrivial.~Then~the~system~is|+GFcen-F?6#%V(b)~c onsistent~only~if~b~is~in~the~column~space~of~A|+GFienF\fnF[vC*-F?6#%jnThe~set~ of~all~vectors~(x1,x2,x3)~that~satisfies~the~equations|+G>82/,(%#x1GFF%#x2GFgv% #x3GFgvF<>83/,(FagnFFFbgnFFFcgnFFF<-Fao6$F^gnFegn-F?6#%en(a)~is~a~subspace~of~t he~3-space~spanned~by~one~vector~R3|+G-F?6#%gn(b)~is~a~subspace~of~the~3-space~ R3~~spanned~by~two~vectors|+G-F?6#%in(c)~is~a~subspace~of~the~3-space~R3~~spann ed~by~three~vectors|+G-F?6#%I(d)~is~not~a~subspace~of~the~3-space~R3|+GF^wC*Fjf n>F^gnF_gn>Fegn/,(FagnF[oFbgnF_]lFcgnF_]lFF^gn/F`gnFF>Fegn/FggnFFFhgnF[inF^inFainFchnFfxC *-F?6#%inThe~set~of~all~vectors~(x1,x2,x3)~that~satisfies~the~equation|+G>81/,( FagnFFFbgnF_]lFcgn!"%%"kG-Fao6#F^jn-F?6#%Dis~a~subspace~of~the~3-space~R3~if|+G -F?6#%)(a)~k=3|+G-F?6#%)(b)~k=2|+G-F?6#%)(c)~k=1|+G-F?6#%)(d)~k=0|+GF\[lC'-F?6# %`oIf~the~zero~vector~does~not~belong~to~a~subset~S~of~a~linear~space~V|+G-F?6# %:(a)~S~is~a~subspace~of~V|+G-F?6#%>(b)~S~is~not~a~subspace~of~V|+G-F?6#%H(c)~S ~is~a~linearly~dependent~set~of~V|+G-F?6#%J(d)~S~is~a~linearly~independent~set~ of~V|+GFa_lC'-F?6#%hnIf~the~zero~vector~belongs~to~a~subset~S~of~a~linear~space ~V|+GFh[oF[\oF^\oFa\oF_alC(-F?6#%enThe~minimum~number~of~vectors~needed~to~span ~the~subspace|+G-F?6#%>S:=|fr(x1,x2,x3)~|gr~x1=x2=0|hr~is|+GFj]n-F?6#%'(b)~2|+G -F?6#%'(c)~1|+G-F?6#%'(d)~0|+GFablC(Fi\o-F?6#%;S:=|fr(x1,x2,x3)~|gr~x1=0|hr~is| +GFj]nF_]oFb]oFe]o/FL"#;C(Fi\o-F?6#%@S:=|fr(x1,x2,x3)~|gr~x1=~x2+x3|hr~is|+GFj] n-F?6#%'(b)~0|+GFb]o-F?6#%'(d)~2|+G/FL"#FC-F?6#%=|+**************************|+G@AFK C'-F?6#%\oIf~S1~is~a~nonempty~subset~of~a~linearly~independent~set~S,~then|+G-F ?6#%A(a)~S1~must~be~a~dependent~set~|+G-F?6#%D(b)~S1~must~be~an~independent~set ~|+G-F?6#%Q(c)~we~cannot~tell~if~S1~is~an~independent~set~|+G-F?6#%N(d)~we~cann ot~tell~if~S1~is~a~dependent~set~|+GFjnC'-F?6#%[oIf~S1~is~a~subset~of~a~set~S~a nd~S1~is~a~independent~set,~then~|+G-F?6#%<(a)~S~must~be~independent~|+G-F?6#%9 (b)~S~may~be~dependent~|+G-F?6#%K(c)~S~can~form~a~basis~for~a~linear~space|+G-F ?6#%io(d)~vectors~in~S~cannot~be~expressed~as~a~linear~combination~of~vectors~i n~S1|+GF]qC'-F?6#%boIf~|frv1,v2,v3|hr~is~a~linearly~independent~set~in~a~linear ~space~V,~then~|+G-F?6#%B(a)~|fr~v1,v2|hr~is~a~dependent~set~|+G-F?6#%H(b)~|frv 1,v1+v2,v3|hr~is~a~independent~set|+G-F?6#%F(c)~|frv1,v1+v2,v2|hr~is~a~dependen t~set|+G-F?6#%I(d)~|frv2,v2+v3,v3|hr~is~an~independent~set|+GF_rC(-F?6#%]oWhile ~checking~if~a~set~of~vectors~in~dependent,~we~ended~up~with|+G-F?6#%]oa~homoge neous~system~of~linear~equations~whose~solution~is~unique|+G-F?6#%N(a)~the~set~ of~vectors~is~linearly~dependent|+G-F?6#%Q(b)~the~set~of~vectors~is~linearly~in dependent~|+G-F?6#%hn(c)~the~set~of~vectors~is~a~not~basis~for~the~solution~spa ce|+G-F?6#%^o(d)~one~of~the~vectors~is~a~linear~combination~of~the~other~vector |+GFasC'Fg_n-F?6#%P(a)~is~an~example~of~a~subspace~of~dimension~1|+G-F?6#%P(b)~ is~an~example~of~a~subspace~of~dimension~2|+GF``nFc`nFctC'-F?6#%`qIf~B=|frv1,~v 2,~v3,..vn|hr~is~a~linearly~independent~set~that~spans~the~linear~space~V,~then ~for~v~in~V~|+G-F?6#%Q(a)~|fr~v,~v1,v2,v3,...,vn|hr~is~linearly~dependent|+G-F? 6#%S(b)~|fr~v,~v1,v2,v3,....vn|hr~is~linearly~independent|+G-F?6#%\o(c)~v~can~b e~expressed~in~many~ways~as~a~linear~combination~of~B|+G-F?6#%`o(d)~v~can~be~ex pressed~in~only~two~ways~as~a~linear~combination~of~B|+GF_uC)-F?6#%jnA~linearly ~independent~set~that~spans~the~rows~of~the~matrix~A|+G>Fdo-Fio6#7%7&FFFF_yFaam>FiyFeamFd_ m-F?6#%P(a)~are~linearly~dependent~if~a=1,b=4,~and~c=4|+G-F?6#%P(b)~are~linearl y~dependent~if~a=2,b=4,~and~c=8|+G-F?6#%P(c)~are~linearly~dependent~if~a=1,b=2, ~and~c=8|+G-F?6#%P(d)~are~linearly~dependent~if~a=2,b=2,~and~c=4|+GF^wC*F\go>F_ yFaam>FiyFeamFd_m-F?6#%U(a)~are~linearly~independent~if~a=-1,b=-2,~and~c=-4|+G- F?6#%R(b)~are~linearly~independent~if~a=2,b=4,~and~c=8|+G-F?6#%S(c)~are~linearl y~independent~if~a=4,b=8,~and~c=16|+G-F?6#%R(d)~are~linearly~independent~if~a=1 ,b=2,~and~c=8|+GFdwC'-F?6#%SWhich~one~of~the~following~statements~is~correct?|+ G-F?6#%U(a)~Every~subset~of~an~independent~set~is~dependent|+G-F?6#%en(b)~Every ~subset~of~a~linearly~dependent~set~is~dependent|+G-F?6#%in(c)~Every~subset~of~ a~linearly~independent~set~is~independent|+G-F?6#%]o(d)~A~dependent~set~can~be~ extended~to~a~linearly~independent~set|+GFfxC'F]io-F?6#%S(a)~The~spanning~set~o f~a~linear~space~is~~unique|+G-F?6#%in(b)~The~spanning~set~forms~always~a~basis ~for~a~linear~spaces|+G-F?6#%]o(c)~A~dependent~set~can~be~extended~to~a~basis~f or~a~linear~space|+G-F?6#%co(d)~An~independent~set~cannot~be~extended~to~a~basi s~for~a~linear~space|+GF\[lC--F?6#%^pThe~number~of~the~independent~vectors~that ~span~the~space~R2~or~a~subsapce~of~R2~isG>F_yF]fm>Fiy-Fcy6#7$F[oF^q>F^hm-Fcy6# 7$FbsFbsF_^mFd^mF_hmFj]n-F?6#%'(b)~1|+GF`^nFe]oFa_lC--F?6#%^pThe~number~of~the~ independent~vectors~that~span~the~space~R3~or~a~subspace~of~R3~isG>F_y-Fcy6#7%F FFgvFF>FiyF\jm>F^hm-Fcy6#7%F^qF^qF`uF_^mFd^mF_hmFj]nFf[p-F?6#%'(c)~0|+GFe^oF_al C(-F?6#%ZThe~number~of~independent~vectors~that~span~the~subspace|+GF_^oFj]nFb^ oFb]oFe^oFablC(Fj\pF[_oFj]nF_]oFb]oF^_oF]clF7F7F7, I/questionset3_5:F$F&F7F7C%@$F:C$F>FCFf_o@EFKC'-F?6#%]oIf~S1~is~a~non-empty~sub set~of~a~linearly~independent~set~S,~then|+GF^`oFa`oFd`oFg`oFjnC'F[aoF^aoFaaoFd aoFgaoF]qC'-F?6#%MIf~a~linear~space~V~is~of~dimension~5,~then|+G-F?6#%M(a)~any~ set~of~5~vectors~in~V~forms~a~basis|+G-F?6#%fn(b)~any~set~of~5~or~less~vectors~ in~V~must~be~independent~|+G-F?6#%en(c)~any~set~of~more~than~5~vectors~in~V~mus t~be~dependent|+G-F?6#%gn(d)~any~set~of~more~than~5~vectors~in~V~must~be~indepe ndent|+GF_rC'F[boF^boFaboFdboFgboFasC(F[coF^coFacoFdcoFgcoFjcoFctC'-F?6#%A~A~li near~space~of~dimension~n~|+G-F?6#%H(a)~has~a~unique~basis~with~n~vectors~|+G-F ?6#%U(b)~can~have~many~bases~with~any~number~of~elements|+G-F?6#%V(c)~can~have~ many~bases~but~all~must~have~n~elements|+G-F?6#%hn(d)~can~have~only~two~bases~w ith~n~elements~in~the~basis~set|+GF_uC'-F?6#%foIf~B=~|frv1,~v2,~v3,...vn|hr~is~ a~basis~for~a~linear~space~V,~then~for~v~in~V~|+GFhdoF[eoF^eoFaeoF[vC'F]cn-F?6# %C(a)~is~a~subspace~of~dimension~6~|+G-F?6#%B(b)~is~a~subspace~of~dimension~5|+ GFfcnFicnF^wC)-F?6#%KA~basis~for~the~row~space~of~the~matrix~A|+G>FdoFieoFgilF_ foFbfo-F?6#%<(c)~~|fr[1,0,1,1],[2,0,2,2]|hr|+GFhfoFdwC'F]io-F?6#%K(a)~The~basis ~of~a~linear~space~is~unique|+GF`jo-F?6#%bo(c)~A~dependent~set~can~be~extended~ to~form~a~basis~for~a~linear~space|+G-F?6#%eo(d)~An~independent~set~can~be~exte nded~to~form~a~basis~for~a~linear~space|+GFfxC'F]ioF_apFcioFfio-F?6#%]o(d)~A~de pendent~set~can~be~extended~to~a~basis~for~a~linear~space|+GF\[lC'F]io-F?6#%M(a )~A~basis~of~a~linear~space~is~not~unique|+GF`joFcjoFfjoFa_lC--F?6#%RThe~dimens ion~of~the~space~spanned~by~the~vectorsG>F_yF]fm>FiyF_[p>F^hmFc[pF_^mFd^mF_hmFj ]nFf[pF`^nFe]oF_alC-Fabp>F_yF^\p>FiyF\jm>F^hmFc\pF_^mFd^mF_hmFj]nFf[pFf\pFe^oFa blC--F?6#%MA~basis~for~the~space~spanned~by~the~vectorsG>F_yF]fm>FiyF_[p>F^hmFc [pF_^mFd^mF_hm-F?6#%;(a)~consists~of~v1~and~v2|+G-F?6#%;(b)~consists~of~v1~and~ v3|+G-F?6#%;(c)~consists~of~v2~and~v3|+G-F?6#%3(d)~all~the~above|+GF\^oC+Fabp>F _yF]fm>Fiy-Fcy6#7$F^qF]wF_^mFd^mFj]nFf[pF`^nFe]oFh^oC'-F?6#%[oThe~dimension~of~ the~subspace~S:=|fr(x1,x2,x3,x4)~|gr~x1=~x2=~0|hr~is|+GFj]nF_]oFb]oF^_oF]clF7F7 F7, I/questionset3_6:F$F&F7F7C%@$F:C$F>FC-F?6#%A|+******************************|+G @AFKC'-F?6#%doA~nonzero~5x4~matrix~A~in~the~echelon~form~has~two~dependent~rows ,~then~|+G-F?6#%L(a)~the~column~space~of~A~has~dimension~4~|+G-F?6#%K(b)~the~co lumn~space~of~A~has~dimension~5|+G-F?6#%K(c)~the~column~space~of~A~has~dimensio n~2|+G-F?6#%K(d)~the~column~space~of~A~has~dimension~3|+GFjnC(F[coF^coFacoFdcoF gcoFjcoF]qC'-F?6#%`pIf~an~nxm~nonhomogeneous~system~of~linear~Ax=b~is~such~that ~rank[A|grb]=~rank[A],~then|+G-F?6#%G(a)~the~system~has~a~unique~solution~|+G-F ?6#%N(b)~the~system~has~infinitely~many~solutions|+G-F?6#%T(c)~we~cannot~decide ~if~the~system~has~a~solution~|+G-F?6#%gn(d)~the~associated~homogeneous~system~ has~a~unique~solution|+GF_rC+-F?6#%hnGiven~the~echelon~form~of~an~augmented~mat rix~of~a~system~of|+G-F?6#%2linear~equations|+G>Fdo-Fio6#7%F\fo7&F8*-Fio6#7%7'FFFFdo -Fio6#7%F\foF]hp7&FFham-Fio6#7%7#FF7#F[o7#FFdoFieoFgilF_f oFbfoF[apFhfoF[vC(F]enF`enFcenFfenFienF\fnF^wC(F`fnFcfnFcenFffnFienF\fnFdwC'F]i oF_apF`joFbapFeapFfxC)-F?6#%JThe~row~vectors~of~the~matrix~A~given~by|+G>FdoF`i lFgil-F?6#%K(a)~are~linear~combination~of~one~another|+G-F?6#%<(b)~are~linearly ~dependent|+G-F?6#%>(c)~are~linearly~independent|+G-F?6#%D(d)~span~a~subspace~o 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#%QType~";"~and~Press~Enter~to~get~the~final~score|+GF^eqFC@$/F]eqFbeqC$-Fao6#% AWARNING~!!!~You~exit~too~quicklyGFC-F?6$%ATotal~number~of~questions~=~~%d|+G,& FgeqFFFgvFF-F?6$%ANumber~of~CORRECT~answers~=~~%d|+GFecq-F?6%%NThe~percentage~o f~correct~answers~=~%.1f%c~|+~G,$*&FecqFFFjhqFgv"$+"%"%GFC@$FaeqC$>FiglFaiq@+1" #!*Figl-Fao6$%MOutstanding~Performance.~Percentage~score~isGFigl1"#!)Figl-Fao6$ %FGood~performance.~Percentage~score~isGFigl1"#qFigl-Fao6$%IAverage~Performance .~Percentage~score~isGFigl1"#gFigl-Fao6$%OBelow~average~Performance.~Percentage ~score~isGFigl-Fao6$%LNot~a~good~performance.~Percentage~score~isGFiglFC@$0,&Fe cqFFFgeqFgvFFhamF^eq@%5/Fham%$yesG/Fham%$YESGC&-Fao6#%?INCORRECTLY~ANSWERED~Q UESTIONSG-Fao6#%?******************************GFC?(FdoFFFF,&FeeqFFFgvFF%%trueG C&-F?6$%6Question~Number~%d~:~G&&F[fqF[jl6#F^q-%/questionset1_2G6#&F]]rFE>F`ipF ^eq@%/F`ip&F]]r6#F[o-Fao6#%AGOOD.~This~is~the~correct~answerGC$-Fao6#%FIncorrec 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testing~system~of~ILAT.|+GFCFjcr-F?6#%GTo~take~another~quiz,~type~quiz3_2();|+G F`dr>FhamF^eqF7Fa_r%(quiz3_6G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fao6#%>Quiz~ on~Row~and~Coulmn~SpacesGF[`sF]dqF`dqFcdqFfdqFCFfcs>F]eqF^eq@$FaeqC7>FeeqFF>Fge qFF>F_yFieq>F[fqF\fq>Fham61F`pF,F,F]pF`pF_pF,F_pF,F`pF_pF]pF_pF]pF_p>FBFF?(F7FF FFF7FbfqC$>Fi\lFffq@$F]gqC%-%,question3_6GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^eqFC@ $FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqFjjqF ][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-FhdpFa]r>F` ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCF^fsFCFjcr-F?6#%GTo~take~another~quiz,~type~qui z3_6();|+GF`dr>FhamF^eqF7Fa_r%(quiz3_4G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fa o6#%GQuiz~on~Linear~Dependence/IndependenceGF[`sF]dqF`dqFcdqFfdqFC-F?6#%inType~ ;~and~press~Enter~to~continue;~or~exit;~to~quit~the~test|+G>F]eqF^eq@$FaeqC7>Fe eqFF>FgeqFF>F_yFieq>F[fqF\fq>Fham61F,F,F,F,F,F]pF,F,F`pF_pF]pF_pF`pF`pF,>FBFF?( F7FFFFF7FbfqC$>Fi\lFffq@$F]gqC%-%,question3_4GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^e qFC@$FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqF jjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-Fa_oFa] r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCF^fsFCFjcr-F?6#%GTo~take~another~quiz,~type ~quiz3_4();|+GF`dr>FhamF^eqF7Fa_r%(quiz3_1G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F <-Fao6#%=~~~~~~~Quiz~on~Linear~SpacesGF[`sF]dqF`dqFcdqFfdqFC-F?6#%[oType~;~and~ press~Enter~to~continue;~or~exit;~to~quit~the~test~>|+G>F]eqF^eq@$FaeqC7>FeeqFF >FgeqFF>F_yFieq>F[fqF\fq>Fham61F]pF,F_pF,F`pF]pF`pF]pF`pF,F]pF_pF`pF_pF]p>FBFF? (F7FFFFF7FbfqC$>Fi\lFffq@$F]gqC%-%,question3_1GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^ eqFC@$FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjq FjjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-F"Fa]r >F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCF^fsFCFg^r-F?6#%I~~To~take~another~quiz,~typ e~quiz3_1();|+GF]_r>FhamF^eqF7Fa_r%(quiz3_3G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F; F<-Fao6#%2Quiz~on~SubspacesGF[`sF]dqF`dqFcdqFfdqFCF`js>F]eqF^eq@$FaeqC7>FeeqFF> FgeqFF>F_yFieq>F[fqFe`s>Fham63F,F,F`pF_pF,F_pF,F]pF,F`pF`pF,F_pF_pF,F`pF,>FBFF? (F7FFFFF7FbfqC$>Fi\lF]as@$F]gqC%-%,question3_3GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^ eqFC@$FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjq FjjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-F]_nFa ]r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFhbsF^fsFCFjcr-F?6#%GTo~take~another~quiz,~t ype~quiz3_3();|+GF`dr>FhamF^eqF7Fa_rF7, I,question3_1:6$%(quesnumGFjbq6#%'count1GF7F7C*-F"6#FL-F?6#%XEnter~your~answer~ below~OR~type~exit;~to~quit~the~test|+GFC>Fgeq,&FgeqFFFFFFFC>F]eqF^eq>FdoFF@$Fa eq@%/9%F]eqC$-Fao6#%FVERY~GOOD.~This~is~the~correct~answerG>Fecq,&FecqFFFFFF@$/ FdoFFC%-F?6#%6Incorrect.~Try~again|+G>F]eqF^eq@%FedtC$-Fao6#%GVERY~GOOD.~This~i s~~the~correct~answerG>FecqF\etC'-F?6#%5Sorry.~Wrong~Again!|+GFC>&F[fq6#Feeq6%F LFfdtFB>Feeq,&FeeqFFFFFF>Fdo,&FdoFFFFFFF76'FgeqFecqF[fqFeeqF]eqF7, I,question3_2:FectFgctF7F7C*-F`clF[dtFCF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fe dtC$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Fe eqFcft>FdoFeftF7FfftF7, I/questionset2_1:F$6,F'F(F)F*F+%#B1G%#B2G%#B3G%#B4GF-F7F7C&@$F:C$F>FC-F?6#%Y|+* *****************************************************|+G@AFKC+-F?6#%.The~matric es|+G>Fdo-Fio6#7#7%FF,&FevFFFfvFFFhv>Fiy-Fio6#7#7%FevFhvFdt-Fao6$FiilFf^m-F?6#% .are~equal~if|+G-F?6#%/(a)~x=1,y=z=6|+G-F?6#%1(b)~x=1,y=5,z=6|+G-F?6#%2(c)~x=1, y=-5,z=6|+G-F?6#%1(d)~x=1,y=z,z=5|+GFjnC'-F?6#%QThe~product~of~two~upper~triang ular~matrices~is|+G-F?6#%@(a)~an~upper~triangular~matrix|+G-F?6#%?(b)~a~lower~t riangular~matrix|+G-F?6#%9(c)~a~symmetric~matrix~|+G-F?6#%8(d)~a~diagonal~matri x~|+GF]qC+-F?6#%XThe~echelon~form~of~an~augmented~matrix~of~a~system~of|+G-F?6# %>linear~equations~is~given~by|+G>Fdo-Fio6#7%7&FFFFdo.Fdo-Fao6$%(That~isG/*$FdoF^qFdo-F?6#%&The n|+G-F?6#%C(a)~A~can~only~be~the~zero~matrix|+G-F?6#%G(b)~A~can~only~be~the~ide ntity~matrix|+G-F?6#%?(c)~A~can~only~be~invertible|+~G-F?6#%<(d)~A~to~the~power ~15~is~A|+GFasC*F[it>Fdo-Fio6#7$Fbit7%F[o,&FevFFFfvFgvF`r>Fiy-Fio6#7$Fhit7%,&Fh vFFFajnFFFajnF`rFiit-F?6#%<(a)~are~equal~if~x=1,y=z=6|+G-F?6#%>(b)~are~equal~if ~x=1,y=5,z=6|+G-F?6#%?(c)~are~equal~if~x=1,y=-5,z=6|+G-F?6#%P(d)~are~not~equal~ for~any~values~of~x,y,~and~z|+GFctC'-F?6#%TThe~product~of~an~nxr~matrix~A~and~r xm~matrix~B~is|+G-F?6#%3(a)~an~mxr~matrix|+G-F?6#%3(b)~an~nxm~matrix|+G-F?6#%1( c)~not~defined|+G-F?6#%0(d)~rxn~matrix|+GF_uC)-F?6#%TThe~(3,2)~entry~in~the~pro duct~of~the~two~matrices|+G-Fao6#/Fdo-Ffo6#-Fio6#7%7%FFF[oF<7%F]wF[oFF7%F^qFFF^ q-Fao6#/Fiy-Ffo6#-Fio6#7%7$FFF[o7$F]wF[o7$FFFbs-F?6#%,(a)~is~-23|+G-F?6#%+(b)~i s~23|+G-F?6#%*(c)~is~6|+G-F?6#%+(d)~is~15|+GF[vC'F]io-F?6#%R(a)~The~sum~of~two~ nxm~matrices~is~an~nxm~matrix|+G-F?6#%U(b)~The~product~of~any~two~matrices~is~w ell~defined|+G-F?6#%[o(c)~Multiplication~of~matrices,~whenever~defined,is~commu tative|+G-F?6#%hn(d)~Addition~of~matrices~whenever~defined~is~not~commutative|+ GF^wC'-F?6#%WWhich~one~of~the~following~statements~is~not~correct?|+GFfdu-F?6#% ^o(b)~The~product~of~an~nxm~matrix~and~an~mxn~matrix~is~well~defined|+G-F?6#%\o (c)~Multiplication~of~matrices,~whenever~defined,~is~commutative|+G-F?6#%Z(d)~A ddition~of~matrices~whenever~defined~is~commutative|+GFdwC'F]io-F?6#%jn(a)~Addi tion~of~matrices,~whenever~defined,~is~not~associative|+G-F?6#%T(b)~The~additiv e~inverse~of~a~matrix~is~not~unique|+G-F?6#%N(c)~The~zero~matrix~is~the~additiv e~identity|+GF_euFfxC'F]io-F?6#%in(a)~The~product~of~a~matrix~with~a~scalar~is~ always~a~scalar~|+G-F?6#%^o(b)~The~product~of~a~matrix~with~a~scalar~is~a~matri x~of~same~size|+G-F?6#%[o(c)~The~product~of~a~matrix~with~a~scalar~is~not~alway s~defined|+G-F?6#%\o(d)~The~product~of~a~matrix~A~with~a~scalar~is~always~equal ~to~A|+GF\[lC*-F?6#%,The~matrix|+G-Fao6#/Fdo-Ffo6#-Fio6#7$7$FFFF7$FgvFF-F?6#%:c ommutes~with~the~matrix|+G-Fao6#/Fiy-Ffo6#-Fio6#7$F\p7$FgvF,-F?6#%G(a)~if,~for~ example,~a~=~2~~and~b~=~3|+G-F?6#%H(b)~if,~for~example,~a~=~3~~and~b~=~3~|+G-F? 6#%G(c)~if,~for~example,~a~=~3~~and~b~=~2|+G-F?6#%G(d)~if,~for~example,~a~=~4~a nd~~b~=~5|+GFa_lC*-F?6#%EThe~value~of~a~for~which~the~matrix|+G-Fao6#/Fdo-Ffo6# -Fio6#7$F\pFbhu-F?6#%?is~a~solution~to~the~equation|+G-Fao6#/,&*$FevF[oFFFevF_] lF<-F?6#%+(a)~is~-1|+G-F?6#%*(b)~is~1|+G-F?6#%*(c)~is~2|+G-F?6#%+(d)~is~-2|+GF_ alC)-F?6#%2Given~the~matrix|+G-Fao6#/Fdo-Ffo6#-Fio6#7$F_fmFchu-F?6#%8The~5-th~p ower~of~A~is|+G-F?6#%;(a)~2~times~the~matrix~A~|+G-F?6#%;(b)~16~times~the~matri x~A|+G-F?6#%;(c)~32~times~the~matrix~A|+G-F?6#%;(d)~10~times~the~matrix~A|+GFab lC,F]\vF`\v-F?6#%P(a)~The~matrix~A~is~a~solution~to~the~equation|+G-Fao6#/,&*$F evF^qFFFev!#;F<-F?6#%P(b)~The~matrix~A~is~a~solution~to~the~equation|+G-Fao6#/, &F_^vFFFevF_]lF<-F?6#%P(c)~The~matrix~A~is~a~solution~to~the~equation|+G-Fao6#/ ,&F_^vFFFevFajnF<-F?6#%P(d)~The~matrix~A~is~a~solution~to~the~equation|+G-Fao6# /,&F_^vFFFev!")F<-Fao6#%AAdd~more~questions~to~this~test!G-F?6#%G|+************ ************************|+GF7F7F7, I,question3_3:FectFgctF7F7C*-F]_nF[dtFCF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fe dtC$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFhbs>F_ftFaft> FeeqFcft>FdoFeftF7FfftF7, I/questionset2_2:F$F^htF7F7C&@$F:C$F>FCFfht@AFKC'F[[uF^[u-F?6#%:(b)~an~invertib le~matrix|+GFd[uFg[uFjnC+F[\uF^\u>FdoFb\uFgilF]]uF`]uFc]uFf]uFi]uF]qC*F[\uF^\u> Fdo-Fio6#7%Fe\uFg\u7&FFdo-Fio6#7%F[hm7%FFdo-Fio6#7%7%FFFFdo-Fio6#7%F__mF[amFicvFgil-F?6#%2is~an~example~of|+G-F?6#%:(a)~an~ele mentary~matrix|+G-F?6#%7(b)~a~diagonal~matrix|+G-F?6#%J(c)~a~matrix~in~reduced~ row~echelon~form|+G-F?6#%6(d)~symmetrix~matrix|+GF_uC*Facv>Fdo-Fio6#7&7&FFFFFFdoFh_u>FiyF^`uFiitFc`uFf`uFi`uF\auF^wC)Facv>Fdo-Fio6#7%Fg`m7%FFdo-Fio6#7%F[hmF[amFicvFgil-F?6#%;then~the~matrix~A^(10)~is|+G>F[elFihv>Fbel -Fio6#7%F``mF[amFicv>Fhel-Fio6#7%7%FFFFigl-Fio6#7%Fg`mF[amFicv-Fao 6$%$(a)GFbel-Fao6$%$(b)GF[el-Fao6$%$(c)GFhel-Fao6$%$(d)GFiglF\[lC'-F?6#%PWhich~ one~of~the~following~statements~is~true?|+G-F?6#%T(a)~Any~two~matrices~always~c ommute.~That~is~AB=BA|+G-F?6#%`o(b)~The~product~of~two~upper~triangular~matrice s~is~upper~triangular|+G-F?6#%\o(c)~Every~symmetric~matrix~is~equivalent~to~an~ elementary~matrix|+G-F?6#%gn(d)~The~transpose~of~a~symmetric~matrix~A~is~not~eq ual~to~A|+GFa_lC)-F?6#%^oA~is~an~nxn~matrix~which~is~not~row~equivalent~to~the~ identity.~If|+G-F?6#%]oA~is~the~coefficient~matrix~of~a~homogeneous~system~of~e quations,|+G-F?6#%5then~the~system~has|+G-F?6#%1(a)~no~solution|+G-F?6#%7(b)~a~ unique~solution|+G-F?6#%?(c)~infinitely~many~solutions|+GFcxF_alC)-F?6#%\oA~is~ an~nxn~matrix~which~is~row~equivalent~to~the~identity.~If~A|+G-F?6#%[ois~the~co efficient~matrix~of~a~homogeneous~system~of~equations,|+GF`\wFc\wFf\wFi\wFcxFab lC)-F?6#%YPre-multiplying~a~2x2~matrix~A~by~the~elementary~matrix|+G-Fao6#/%"EG -Ffo6#-Fio6#7$F_el7$F^qFF-F?6#%2is~equivalent~to|+G-F?6#%A(a)~multiplying~row~2 ~of~A~by~3|+G-F?6#%U(b)~multiplying~row~1~of~A~by~3~and~adding~to~row~2|+G-F?6# %A(c)~multiplying~row~1~of~A~by~3|+G-F?6#%E(d)~adding~row~1~of~A~to~row~2~of~A| +GFg_vFj_vF7F7F7, I,question3_4:FectFgctF7F7C*-Fa_oF[dtFCF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fe dtC$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Fe eqFcft>FdoFeftF7FfftF7, I/questionset2_3:F$F^htF7F7C&@$F:C$F>FCFfht@KFKC'-F?6#%RThe~product~of~two~inve rtible~matrices~is~always|+G-F?6#%4(a)~not~invertible|+G-F?6#%0(b)~invertible|+ G-F?6#%7(c)~a~diagonal~matrix|+G-F?6#%9(d)~a~triangular~matrix|+GFjnC'-F?6#%boI f~the~square~of~a~non~identity~nxn~matrix~A~is~itself,~then~A~must~be|+G-F?6#%0 (a)~invertible|+G-F?6#%.(b)~identity|+G-F?6#%.(c)~singular|+G-F?6#%-(d)~a~or~c~ |+GF]qC(-F?6#%jnIf~the~coefficient~matrix~of~a~nonhomogeneous~system~of~linear| +G-F?6#%Pequations~is~invertible,~then~the~solution~set|+G-F?6#%>(a)~consists~o f~two~elements|+G-F?6#%1(b)~is~infinite|+G-F?6#%<(c)~consist~of~one~element|+G- F?6#%/(d)~is~empty~|+GF_rC(F\cw-F?6#%Nequations~is~singular,~then~the~solution~ set|+G-F?6#%/(a)~is~empty~|+GFecwFhcw-F?6#%8(d)~none~of~the~above~|+GFasC'-F?6# %O~The~sum~of~two~invertible~matrices~is~always|+GF_bw-F?6#%.(b)~singular|+G-F? 6#%0(c)~triangular|+GFcxFctC)Facv>FdoFecvFgilFjcvF]dv-F?6#%H(c)~is~an~example~o f~a~singular~matrix|+G-F?6#%K(d)~is~an~example~of~an~invertible~matrix|+GF_uC'- F?6#%YIf~a~matrix~A~is~such~that~A~is~equal~to~its~cube,~then|+G-F?6#%?(a)~A~mu st~be~the~zero~matrix|+G-F?6#%B(b)~A~must~be~the~identity~matrixG-F?6#%;(c)~A~m ust~be~invertible|+~GFc_uF[vC'-F?6#%FIf~a~matrix~A~has~an~inverse~B,~then|+G-F? 6#%W(a)~B~is~not~unique~since~A~has~more~than~one~inverse|+G-F?6#%T(b)~B~is~not ~row~equivalent~to~the~identity~matrix|+G-F?6#%P(c)~B~is~row~equivalent~to~the~ identity~matrix|+G-F?6#%B(d)~(a)~and~(b)~are~both~correct|+GF^wC'-F?6#%VIf~a~ma trix~A~is~row~equivalent~to~a~matrix~B,~then~|+G-F?6#%E(a)~B~is~invertible~whil e~A~is~not~|+G-F?6#%I(b)~A~and~B~donot~share~same~properties|+G-F?6#%G(c)~A~and ~B~do~share~same~properties~|+G-F?6#%O(d)~A~is~invertible~while~B~is~not~invert ible|+GFdwC*Facv>FdoFaevFgilFdevFgevFjevF]fvF`fvFfxC*Facv>FdoFefvFgilFdevFgevFj evFjfvF`fvF\[lC'-F?6#%HThe~inverse~of~an~elementary~matrix~is|+G-F?6#%A(a)~is~n ot~an~elementary~matrix|+G-F?6#%=(b)~is~an~elementary~matrix|+G-F?6#%<(c)~is~a~ triangular~matrix|+G-F?6#%;(d)~is~a~symmetrix~matrix|+GFa_lC1-F?6#%EThe~inverse ~of~the~diagonal~matrix:|+G>Fdo-%%diagG6'FFF[oF]wF`rFj[lFgil-F?6#%<(a)~is~the~d iagonal~matrix|+G>Fham-Fajw6'FgvF_]lF^qFajnFbs-FaoFgjp-F?6#%<(b)~is~the~diagona l~matrix|+G>F`ip-Fajw6'Fgv#FgvF[o#FFF^q#FgvF`r#FFFbs-FaoF\[q-F?6#%<(c)~is~the~d iagonal~matrix|+G>Fi\l-Fajw6'FF#FFF[o#FgvF^q#FFF`r#FgvFbs-FaoFhem-F?6#%<(d)~is~ the~diagonal~matrix|+G>F_y-Fajw6'FFF[\xFa[xF]\xFc[x-FaoFc^mF_alC'-F?6#%QWhich~o ne~of~the~following~statements~is~false?|+G-F?6#%Z(a)~The~product~of~two~invert ible~matrices~is~invertible|+G-F?6#%en(b)~The~inverse~of~a~diagonal~matrix~is~a ~diagonal~matrix|+G-F?6#%N(c)~Every~invertible~matrix~has~two~inverses|+G-F?6#% T(d)~The~inverse~of~a~symmetric~matrix~is~symmetric|+GFablC'Fjjv-F?6#%P(a)~The~ product~of~two~matrices~is~commutative|+G-F?6#%L(b)~The~inverse~of~a~matrix~is~ ~not~unique|+G-F?6#%S(c)~The~transpose~of~a~symmetric~matrix~is~itself|+G-F?6#% X(d)~The~inverse~of~a~symmetric~matrix~is~not~symmetric|+GF\^oC'-F?6#%RWhich~on e~of~the~following~statements~is~correct|+G-F?6#%^o(a)~The~product~of~a~singula r~and~a~nonsingular~matrix~is~singular|+G-F?6#%ao(b)~The~product~of~a~singular~ and~a~nonsingular~matrix~is~nonsingular|+G-F?6#%Y(c)~The~product~of~two~singula r~matrices~is~nonsingular|+G-F?6#%Y(d)~The~product~of~two~nonsingular~matrices~ is~singular|+GFh^oC(-F?6#%^oGiven~an~nxn~matrix~A~which~is~not~row~equivalent~t o~the~identity,|+G-F?6#%6then~the~matrix~A~is|+G-F?6#%1(a)~nonsingular|+GF\ew-F ?6#%/(c)~symmetric|+GFcx/FL"#=C)-F?6#%^oGiven~an~nxn~matrix~A~which~is~not~row~ equivalent~to~the~identity.|+G-F?6#%`oIf~A~is~the~coefficient~matrix~of~a~homog eneous~system~of~equations,|+GF`\wFc\wFf\wFi\wFcx/FL"#>C)-F?6#%jnGiven~an~nxn~m atrix~A~which~is~row~equivalent~to~the~identity.|+GFg`xF`\wFc\wFf\wFi\wFcx/FLFh `rC)F]ax-F?6#%hnIf~A~is~the~coefficient~matrix~of~a~nonhomogeneous~system~of|+G -F?6#%@equations,~then~the~system~has|+GFc\wFf\wFi\wFcxFg_vFj_vF7F7F7, I,question3_5:FectFgctF7F7C)-F^]pF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Feeq Fcft>FdoFeftF7FfftF7, I,question3_6:FectFgctF7F7C*-FhdpF[dtFCF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fe dtC$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFhbs>F_ftFaft> FeeqFcft>FdoFeftF7FfftF7, I/questionset2_4:F$F^htF7F7C&@$F:C$F>FC-F?6#%F|+******************************* ****|+G@AFKC)-F?6#%]oThe~coefficient~matrix~of~a~nonhomogeneous~system~of~equat ions~is|+G-Fao6#/Fdo-Ffo6#-Fio6#7%F__mFhcvF[hm-F?6#%Ethen~the~solution~set~of~t he~system|+GFbcwFecwFhcwF[dwFjnC)F\ex-Fao6#/Fdo-Ffo6#-Fio6#7%F__mFhcv7%FgvFFdoF`jwFgilFcjw>FhamFgjwFijwFjjw>F`ipF^[xFd[xFe[x>Fi\lFi[xF_\ xF`\x>F_yFd\xFf\xFdwC'Fh\xF[]xF^]xFa]xFd]xFfxC)Fd`xFg`xF`\wFc\wFf\wFi\wFcxF\[lC )F]axFg`xF`\wFc\wFf\wFi\wFcxFa_lC'-F?6#%WThe~inverse(AB)~of~two~invertible~matr ices~A~and~B~is|+G-F?6#%>(a)~is~inverse(A)*inverse(B)|+G-F?6#%>(b)~is~inverse(B )*inverse(A)|+G-F?6#%4(c)~is~inverse(BA)|+G-F?6#%4(d)~does~not~exist|+GF_alC(Fg gu-Fao6#/Fdo-Ffo6#-Fio6#7%F`\p7%F[oFFF^q7%F`rFgvFbs-F?6#%H(a)~is~an~example~of~ a~singular~matrix|+G-F?6#%J(b)~is~an~example~of~a~elementary~matrix|+G-F?6#%K(c )~is~an~example~of~a~nonsingular~matrix|+G-F?6#%I(d)~is~an~example~of~a~symmetr ic~matrix|+GFablC.-F?6#%4Given~the~matrices|+G-Fao6#/Fdo-Ffo6#-Fio6#7$FbhuFfel- Fao6#/Fiy-Ffo6#-Fio6#7$F_elFbhu-F?6#%BThe~inverse~of~the~product~AB~is|+G-F?6#% 0(a)~the~matrix|+G-Fao6#/%"CG-Ffo6#-Fio6#7$F_fm7$FgvF[o-F?6#%0(b)~the~matrix|+G -Fao6#/Fe]y-Ffo6#-Fio6#7$FbhuFfcu-F?6#%0(c)~the~matrix|+G-Fao6#/Fe]y-Ffo6#-Fio6 #7$F_fmFfcu-F?6#%0(d)~the~matrix|+G-Fao6#/Fe]y-Ffo6#-Fio6#7$FbhuF[^yFg_vFj_vF7F 7F7, I/questionset2_5:F$F^htF7F7C&@$F:C$F>FCFfht@KFKC(-F?6#%^oThe~determinant~of~a~4 x4~triangular~matrix~whose~diagonal~elements|+G-F?6#%6are~-2,~3,1~and~2~is|+G-F ?6#%'(a)~4|+G-F?6#%((b)~-6|+G-F?6#%((c)~12|+G-F?6#%*(d)~-12~|+GFjnC)-F?6#%KThe~ determinant~of~the~matrix~A~given~by:|+G>Fdo-Fio6#7&7&F[oFFFFFbs7&FFdo-Fio6#7 %F[hmFhcv7%FFdo-Fio6#7$7$F]pF ,F^pFgil-F?6#%^othe~cofactor~of~each~entry~is~equal~to~the~entry~itself,~then~A ~is|+G>F[el-Fio6#7$Feey7$,$F,FgvF]p>Fbel-Fio6#7$7$F]p,$F_pFgv7$F^fyF`p>FhelFbey >Figl-Fio6#7$7$F`pFdfyF]fy-F?6#%.(a)~given~by|+G-FaoF`fl-F?6#%.(b)~given~by|+G- FaoFefl-F?6#%.(c)~given~by|+G-FaoFjfl-F?6#%.(d)~given~by|+G-FaoFahlFctC)-F?6#%j nThe~values~of~x~for~which~the~following~matrix~is~singular~are|+G>Fdo-Fio6#7%7 %FevFFF<7%F(a)~is~singular~are~|fr1,-1/2|hr|+G-F?6#% ?(b)~is~singular~are~|fr-1,-1/2|hr|+G-F?6#%>(c)~is~singular~are~|fr-1,1/2|hr|+G -F?6#%=(d)~is~singular~are~|fr1,1/2|hr|+GF_uC'-F?6#%aoIf~A~is~2x2~invertible~ma trix,~then~the~determinant~of~the~adjoint(A)|+G-F?6#%:(a)~is~equal~to~1/det(A)| +G-F?6#%9(b)~is~equal~to~-det(A)|+G-F?6#%8(c)~is~equal~to~det(A)|+G-F?6#%4(d)~i s~equal~to~-1|+GF[vC(-F?6#%_oIf~the~determinant~of~a~4x4~matrix~A~is~-3,~then~t he~determinant~of|+G-F?6#%+of~2*A~is|+G-F?6#%((a)~-6|+G-F?6#%'(b)~6|+G-F?6#%((c )~48|+G-F?6#%)(d)~-48|+GF^wC(-F?6#%_oIf~a~6x6~matrix~A~is~such~that~one~row~is~ a~multiple~of~another~row|+G-F?6#%Fthen~the~determinant~of~the~matrix~A|+G-F?6# %'(a)~6|+GFf[pFf\p-F?6#%((d)~-6|+GFdwC(-F?6#%^oIf~the~determinant~of~a~matrix~A ~is~0,~then~the~homogeneous~system|+G-F?6#%Sof~linear~equations~whose~coefficie nt~matrix~is~A|+G-F?6#%B(a)~has~infinitely~many~solution|+G-F?6#%F(b)~has~a~uni que~nontrivial~solution|+G-F?6#%C(c)~has~a~unique~trivial~solution|+G-F?6#%5(d) ~is~inconsistent|+GFfxC(-F?6#%jnIf~the~determinant~of~a~matrix~A~is~0,~then~the ~nonhomogeneous|+G-F?6#%insystem~of~linear~equations~Ax=b~whose~coefficient~mat rix~is~A|+GFj\zF]]z-F?6#%L(c)~may~be~consistent~for~some~values~of~b|+G-F?6#%<( d)~is~always~inconsistent|+GF\[lC(-F?6#%[oIf~the~determinant~of~a~matrix~A~is~n ot~0,~then~the~homogeneous|+G-F?6#%Zsystem~of~linear~equations~whose~coefficien t~matrix~is~A|+GFj\zF]]zF`]zFc]zFa_lC(-F?6#%^oIf~the~determinant~of~a~matrix~A~ is~not~0,~then~the~nonhomogeneous|+GFj]zFj\z-F?6#%E(b)~has~a~unique~solution~fo r~any~b|+GF]^zF`^zF_alC(-F?6#%gnIf~B~is~an~nxn~matrix~obtained~from~a~matrix~A~ by~replacing|+G-F?6#%Othe~i-th~row~of~A~with~2*row1+row2~of~A,~then|+G-F?6#%5(a )~det(B)=2*det(A)|+G-F?6#%4(b)~det(B)=-det(A)|+G-F?6#%6(c)~det(B)=-2*det(A)|+G- F?6#%3(d)~det(B)=det(A)|+GFablC(-F?6#%_oIf~the~second~row~of~a~5x5~matrix~A~is~ a~multiple~of~the~third~row,|+G-F?6#%Rthen~the~determinant~of~the~matrix~A~is~e qual~to|+G-F?6#%((a)~~0|+GFgdyFjcy-F?6#%((d)~~5|+GF\^oC)-F?6#%[oA~matrix~B~is~o btained~from~a~6x6~matrix~A~by~interchanging~the|+G-F?6#%]othird~row~and~the~fi fth~row~of~the~matrix~A.~If~det(B)~is~5,~then|+G-F?6#%Mthe~determinant~of~the~m atrix~A~is~equal~to|+G-F?6#%)(a)~-30|+G-F?6#%((b)~-5|+G-F?6#%)(c)~~30|+GF^azFh^ oC)-F?6#%]oA~matrix~B~is~obtained~from~a~4x4~matrix~A~by~adding~twice~of~the|+G -F?6#%]othird~row~to~the~fourth~row~of~the~matrix~A.~If~det(B)~is~4,~then|+GFha z-F?6#%((a)~~8|+G-F?6#%((b)~-8|+G-F?6#%((c)~-4|+G-F?6#%((d)~~4|+GFa`xC(-F?6#%jo The~diagonal~elements~of~a~5x5~upper~triangular~matrix~A~are:~-1,2,0,4,5.~Then| +G-F?6#%Ethe~determinant~of~the~matrix~A~is:|+G-F?6#%)(a)~~40|+G-F?6#%((b)~~0|+ G-F?6#%((c)-40|+GFcxFj`xC(-F?6#%jnGiven~two~4x4~matrices~A~and~B.~If~the~det(A) =5~and~det(AB)=10|+G-F?6#%Ithen~the~determinant~of~the~matrix~B~is|+G-F?6#%((a) ~~5|+G-F?6#%((b)~-2|+G-F?6#%((c)~-5|+G-F?6#%((d)~~2|+GF`axC(-F?6#%]oGiven~two~5 x5~matrices~A~and~B.~If~the~det(A)=2~and~the~det(AB)=0|+G-F?6#%6then~the~matrix ~B~is|+GF[`x-F?6#%D(b)~row~equivalent~to~the~identity|+GFebwFcxFg_vFj_vF7F7F7, I/questionset2_6:F$F^htF7F7C&@$F:C$F>FCFfht@AFKC(Fd`yFg`yFj`yF]ayF`ayFcayFjnC)F \cy>FdoF`cyFgilFdcyFgcyFjcyF]dyF]qC(FadyFddyFj`yFf[p-F?6#%((c)~-1|+GFjdyF_rC)Fa cv>Fdo-Fio6#7%Fg`mFegvFicvFgilFjcvF]dvFdewFcdvFasC2F^ey>FdoFbeyFgil-F?6#%iothe~ cofactor~of~each~entry~is~equal~to~the~entry~itself,~then~the~matrix~A~is|+G>F[ elFjey>FbelF`fy>FhelFbey>FiglFhfyF\gyF\flF`gyFaflFdgyFfflFhgyF]hlFctC'-F?6#%doI f~A~is~an~invertible~2x2~matrix,~then~the~determinant~of~the~adjoint(A)|+GFgiyF jiyF]jyF`jyF_uC,-F?6#%;The~adjoint~of~the~matrix|+G-Fao6#/Fdo-Ffo6#Fihv-F?6#%3( a)~is~the~matrix|+G-Fao6#/%$AdjGFjhz-F?6#%3(b)~is~the~matrix|+G-Fao6#/Fbiz-Ffo6 #-Fio6#7%F[hmF[amFe]n-F?6#%3(c)~is~the~matrix|+G-Fao6#/Fbiz-Ffo6#-Fio6#7%F[hm7% FFCFG@AFKC'-F?6#%jnThe~LU-decomposition~of~ a~matrix~A~consists~of~the~product~of~|+G-F?6#%C(a)~two~lower~triangular~matric es|+G-F?6#%C(b)~two~upper~triangular~matrices|+G-F?6#%N(c)~an~upper~and~a~lower ~triangular~matrices|+G-F?6#%Q(d)~an~upper~triangular~and~a~diagonal~matrices|+ GFjnC.FjclFjgu-F?6#%XThe~upper~triangular~matrix~in~its~LU-decomposition~is|+G- Fao6#/%"UG-Ffo6#-Fio6#7$Fbhu7$F(b)~The~matrix~ A~is~singular|+G-F?6#%>(c)~The~matrix~A~is~diagonal|+G-F?6#%@(d)~The~matrix~A~i s~triangular|+GF[vC)Fgg\lF\\\lF\^\l-F?6#%7(a)~Then~det(A)~is~-1|+G-F?6#%6(b)~Th en~det(A)~is~1|+G-F?6#%6(c)~Then~det(A)~is~2|+G-F?6#%7(d)~Then~det(A)~is~-2|+GF ^wC-Fjcl-Fao6#/FdoFb^y-F?6#%hnThe~lower~triangular~matrix~in~its~LU-decompositi on,~if~any,|+G-F?6#%B(a)~is~the~inverse~of~the~matrix|+G-Fao6#/FiyFd[\l-F?6#%B( b)~is~the~inverse~of~the~matrix|+G-Fao6#/Fiy-Ffo6#-Fio6#7$Fbhu7$FFF_]l-F?6#%B(c )~is~the~inverse~of~the~matrix|+G-Fao6#/Fiy-Ffo6#-Fio6#7$Fbhu7$FFF][l-F?6#%B(d) ~is~the~inverse~of~the~matrix|+G-Fao6#/FiyFd_\lFdwC-Fjcl-Fao6#/Fdo-Ffo6#-Fio6#7 $7$F`rFF7$F^qF[o-F?6#%UThe~lower~triangular~matrix~in~its~LU-decomposition|+GFj i\l-Fao6#/Fiy-Ffo6#-Fio6#7$7$F`rF<7$F]wF`rF`j\l-Fao6#/Fiy-Ffo6#-Fio6#7$7$Fb[xF< 7$#F^qF`rFFF\[]l-Fao6#/Fiy-Ffo6#-Fio6#7$7$F]\xF<7$#F]wF`rFFFh[]l-Fao6#/FiyFb\]l FfxC-Fjcl-Fao6#/Fdo-Ffo6#-Fio6#7$FbhuFh\]lFi\]lFji\l-Fao6#/Fiy-Ffo6#-Fio6#7$Fch uF`^wF`j\l-Fao6#/FiyFc_]lF\[]l-Fao6#/FiyF[^wFh[]l-Fao6#/Fiy-Ffo6#-Fio6#7$F_el7$ F]wFFF\[lC'-F?6#%OWhich~of~the~following~statements~is~correct?|+G-F?6#%P(a)~Ev ery~matrix~has~a~unique~LU-decomposition|+G-F?6#%O(b)~Some~matrices~have~many~L U-decompositions|+G-F?6#%gn(c)~The~lower~triangular~matrix,~L,~in~A~=~LU~is~inv ertible|+G-F?6#%en(d)~The~lower~triangular~matrix,~L,~in~A~=~LU~is~singular|+GF a_lC'-F?6#%MIf~a~matrix~A~has~an~LU-decomposition,~then|+G-F?6#%Z(a)~L~is~the~i nverse~of~a~product~of~elementary~matrices|+G-F?6#%:(b)~L~is~always~singular|+G -F?6#%<(c)~L~is~a~diagonal~matrix|+GFcxF_alC,-F?6#%FGiven~the~system~of~linear~ equations|+G-Fao6#/,(FagnFFFbgnF_]lFcgnFFFF-Fao6#/,(FagnFgvFbgnFFFcgnFgvF[o-Fao 6#/,(FagnFFFbgnF[oFcgnFgvFbs-F?6#%YThe~solution~y~=~[y1,y2,y3]~of~the~problem~L y~=~[1,2,5]|+G-F?6#%_owhere~L~is~the~lower~triangular~matrix~in~the~LU-decompos ition~of~A|+G-F?6#%2(a)~y~=~[-1,3,1]|+G-F?6#%2(b)~y~=~[1,3,-1]|+G-F?6#%1(c)~y~= ~[1,2,5]|+G-F?6#%2(d)~y~=~[-1,2,5]|+GFablC+F]c]l-Fao6#/,&FagnFFFbgnF_]lFbbl-Fao 6#/,&FagnFgvFbgnFF"#@-F?6#%VThe~solution~y~=~[y1,y2]~of~the~problem~Ly~=~[15,21 ]|+GF_d]l-F?6#%1(a)~y~=~[36,15]|+G-F?6#%1(b)~y~=~[15,21]|+G-F?6#%3(c)~y~=~[-57, -36]|+G-F?6#%1(d)~y~=~[15,36]|+GFg_vFj_vF7F7F7, I*lasterror%Zunable~to~read~`d://publish//ilat4//testxt//lsysquiz.txt`GF7, I'maquiz=F7FabqE\[l(%(quiz2_3G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fao6#%JQuiz ~on~Singular~and~Nonsingular~MatricesGF[`sF]dqF`dqFcdqFfdqFCF`js>F]eqF^eq@$Faeq C7>FeeqFF>FgeqFF>F_yFieq>F[fqFf`r>Fham66F,F_pF_pF`pF`pF`pF`pF_pF_pF]pF_pF,F_pF_ pF_pF]pF,F_pF,F,>FBFF?(F7FFFFF7FbfqC$>Fi\lF_ar@$F]gqC%-%,question2_3GFdgq>F_yFg gq>FBF\hq>F;FFFCF^hqF^eqFC@$FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC$>FiglFaiq@+FiiqF[ jqF^jqF`jqFcjqFejqFhjqFjjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rFb\rFC?(FdoFFF FFf\rFg\rC&Fi\r-Fe`wFa]r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCF^fsFCFjcr-F?6#%GTo~ take~another~quiz,~type~quiz2_3();|+GF`dr>FhamF^eqF7Fa_r%(quiz2_2G:F7FebqF7F7C6 F_cq>FhelFew>FecqF<>F;F<-Fao6#%D~~~Quiz~on~Special~Type~of~MatricesGF[`sF]dqF`d qFcdqFfdqFCF`js>F]eqF^eq@$FaeqC7>FeeqFF>FgeqFF>F_yFieq>F[fqF\fq>Fham61F]pF`pF]p F_pF]pF]pF_pF,F,F,F_pF,F_pF,F,>FBFF?(F7FFFFF7FbfqC$>Fi\lFffq@$F]gqC%-%,question 2_2GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^eqFC@$FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC$>Fi glFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqFjjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\r Fb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-FbavFa]r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCF^fsFC Fjcr-F?6#%GTo~take~another~quiz,~type~quiz2_2();|+GF`dr>FhamF^eqF7Fa_r%(quiz2_1 G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fao6#%7Quiz~on~Matrix~AlgebraGF[`sF]dqF` dqFcdqFfdqFCFfcs>F]eqF^eq@$FaeqC7>FeeqFF>FgeqFF>F_yFieq>F[fqF\fq>Fham61F,F]pF`p F`pF,F,F,F`pF_pF_pF,F,F,F,F_p>FBFF?(F7FFFFF7FbfqC$>Fi\lFffq@$F]gqC%-%,question2 _1GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^eqFC@$FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC$>Fig lFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqFjjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rF b\rFC?(FdoFFFFFf\rFg\rC&Fi\r-F\htFa]r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCF^fsFCF jcr-F?6#%GTo~take~another~quiz,~type~quiz2_1();|+GF`dr>FhamF^eqF7Fa_r%(quiz2_7G :F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fao6#%9Quiz~on~LU-DecompositionGF[`sF]dqF `dqFcdqFfdqFCF`js>F]eqF^eq@$FaeqC7>FeeqFF>FgeqFF>F_yFieq>F[fqF\fq>Fham61F_pF]pF ,F]pF,F`pF]pF,F]pF_pF`pF_pF]pF,F`p>FBFF?(F7FFFFF7FbfqC$>Fi\lFffq@$F]gqC%-%,ques tion2_7GFdgq>F_yFggq>FBF\hq>F;FFFCFiarF^eqFC@$FbhqC$FdhqFCFghqF[iqF^iqFC@$FaeqC $>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqFjjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC& F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-Fih[lFa]r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCF ^fsFCFjcr-F?6#%GTo~take~another~quiz,~type~quiz2_7();|+GF`dr>FhamF^eqF7Fa_r%(qu iz2_6G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fao6#%F]eqF^eq@$FaeqC7>FeeqFF>FgeqFF>F_yFieq>F[fqF\fq>Fham 61F`pF,F,F]pF]pF_pF`pF_pF]pF,F]pF,F`pF,F_p>FBFF?(F7FFFFF7FbfqC$>Fi\lFffq@$F]gqC %-%,question2_6GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^eqFC@$FbhqC$FdhqFCFghqF[iqF^iqF C@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqFjjqF][rFC@$Fa[rC%F]cr>FhamF^eq @%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-FdfzFa]r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^r Fa^rFCF^fsFCFjcr-F?6#%GTo~take~another~quiz,~type~quiz2_6();|+GF`dr>FhamF^eqF7F a_r%(quiz2_4G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fao6#%9Quiz~on~Matrix~Invers ionGF[`sF]dqF`dqFcdqFfdqFCFb\s>F]eqF^eq@$FaeqC7>FeeqFF>FgeqFF>F_yFieq>F[fqF\fq> Fham61F_pF`pF,F_pF,F`pF`pF,F_pF_pF_pF,F,F_pF]p>FBFF?(F7FFFFF7FbfqC$>Fi\lFffq@$F ]gqC%-%,question2_4GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^eqFC@$FbhqC$FdhqFCFghqF[iqF ^iqFC@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqFjjqF][rFC@$Fa[rC%F]cr>Fham F^eq@%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-FbdxFa]r>F`ipF^eq@%Fe]rFh]rC$F\^r F_^rFa^rFCF^fsFCFjcr-F?6#%GTo~take~another~quiz,~type~quiz2_4();|+GF`dr>FhamF^e qF7Fa_r%(quiz2_5G:F7FebqF7F7C6F_cq>FhelFew>FecqF<>F;F<-Fao6#%5Quiz~on~Determina ntsGF[`sF]dqF`dqFcdqFfdqFCFb\s>F]eqF^eq@$FaeqC7>FeeqFF>FgeqFF>F_yFieq>F[fqFf`r> Fham66F`pF,F,F_pF]pF]pF_pF`pF_pF]pF_pF_pF,F`pF]pF,F`pF,F`pF_p>FBFF?(F7FFFFF7Fbf qC$>Fi\lF_ar@$F]gqC%-%,question2_5GFdgq>F_yFggq>FBF\hq>F;FFFCF^hqF^eqFC@$FbhqC$ FdhqFCFghqF[iqF^iqFC@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`jqFcjqFejqFhjqFjjqF][rFC@$ Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\rC&Fi\r-F]`yFa]r>F`ipF^eq@ %Fe]rFh]rC$F\^rF_^rFa^rFCF^fsFCFjcr-F?6#%GTo~take~another~quiz,~type~quiz2_5(); |+GF`dr>FhamF^eqF7Fa_rF7, I%quiz:F$62FfbqFgbqFbjnFhbqFibqFjbqF[cqF\cqF]cq%%set1G%%set2G%%set3G%%set4G%%se t5G%$ANSG%'quesetGF7F7C;>Fiy<:FFFbsFdtF_wFb_lF`alF]^oFi^oFb`xF[axFh`r"#J"#K"#L" #M"#N"#O"#W"#X"#Y"#Z"#["#\"#]>F^hm<5F[oF^qF`rF`uFewFgxF][lFbbl"#A"#B"#C"#D"#E"# F"#G"#H"#I"#T"#U>8/<.FbsF\vF_wFge]l"#P"#Q"#R"#S"#V"#^"#`"#e>F[]m<1"#_Fj__l"#a"# b"#c"#dF[`_l"#fFijq"#h"#i"#j"#k"#l"#m>F^jn<2FijqFd`_lFe`_lFf`_lFg`_lFh`_lFi`_l" #n"#o"#pFdjq"#r"#s"#t"#u"#v@,FK>FegnFiyFjn>FegnF^hmF]q>FegnFb__lF_r>FegnF[]mFas >FegnF^jnF_cq>FhelFew>FecqF<>F;F<@,FK-Fao6#%OQuiz~on~Introduction~to~Linear~Tra nsformationsGFjn-Fao6#%TQuiz~on~Kernel~and~Range~of~a~Linear~TransformationGF]q -Fao6#%XQuiz~on~Matrix~Representation~of~Linear~Transformation.GF_rC$-Fao6#%8Qu iz~on~Change~of~BasisG>FhelFewFasC$-Fao6#%?Quiz~on~Similarity~of~MatricesG>Fhel FewF[`sF]dq-F?6#%`oYou~will~be~given~only~two~chances~to~find~the~correct~answe r.~Enter|+G-F?6#%cothe~answer~followed~by~;~and~press~enter.~You~may~quit~the~t est~anytime|+G-F?6#%boby~typing~>exit;~at~the~Maple~prompt.~However~no~score~wi ll~be~given.~|+GFCFidq>F]eqF^eq@$FaeqC4>FeeqFF>FgeqFF>F_yFieq>F[fq-Fcy6#Fca_l>F ham6foF_pF]pF,F_pF`pF]pF]pF`pF]pF,F_pF]pF,F_pF_pF]pF`pF]pF]pF`pF,F_pF_pF`pF_pF] pF,F_pF`pF]pF_pF`pF]pF_pF,F]pF]pF,F`pF_pF_pF`pF]pF,F_pF,F]pF`pF,F]pF,F_pF,F,F]p F,F`pF]pF`pF]pF,F`pF_pF,F,F_pF_pF,F]pF_pF,F]pF]pF_p>F^gn6foFe]yF'F-Fe]y%"DGF'F' Fgd_lF'F-Fe]yF'F-Fe]yFe]yF'Fgd_lF'F'Fgd_lF-Fe]yFe]yFgd_lFe]yF'F-Fe]yFgd_lF'Fe]y Fgd_lF'Fe]yF-F'F'F-Fgd_lFe]yFe]yFgd_lF'F-Fe]yF-F'Fgd_lF-F'F-Fe]yF-F-F'F-Fgd_lF- Fgd_lF'F-Fgd_lFe]yF-F-Fe]yFe]yF-F'Fe]yF-F'F'Fe]y>FBFF?(F7FFFFF7FbfqC$>Fi\l,&-Fh fq6$FjfqFca_lFFFFFF@$-F_gq6$Fi\lFegn@$F]gqC%-%)questionG6%Fi\lFegq&F^gnFhem>F_y Fggq>FBF\hq>F;FFFCFiar>Fb_rF^eqFC@%FbhqC$-Fao6#%YWARNING~!!!~You~exit~too~quick ly.~No~Score~will~be~givenGFCC&FghqF[iqF^iqFC@$FaeqC$>FiglFaiq@+FiiqF[jqF^jqF`j qFcjqFejqFhjqFjjqF][rFC@$Fa[rC%F]cr>FhamF^eq@%Fi[rC&F_\rFb\rFC?(FdoFFFFFf\rFg\r C&Fi\r-%,questionsetGFa]r>F`ipF^eq@%Fe]rFh]rC$F\^rF_^rFa^rFCFigrFCFg^rF]_r>Fham F^eqF7Fa_rF7, I&blank:6#%"tG6#FfbqF7F7?(FdoFFFFFLFg\r-Fao6#%"~GF7F7F7, I,questionset:F$6'F'F(F)F*F+F7F7C&@$F:C$F>FCFecl@ctFKC(-F?6#%jnIf~B=~|frv1,v2,v 3|hr~is~a~basis~for~the~linear~space~R3~and~T~is~a|+G-F?6#%inone-to-one~and~ont o~linear~transformation~from~R3~to~R3,~then|+G-F?6#%F(a)~T(B)~is~a~linearly~dep endent~set|+G-F?6#%A(b)~T(B)~is~not~a~basis~for~R3~|+G-F?6#%<(c)~T(B)~is~a~basi s~for~R3|+G-F?6#%;(d)~T(B)~does~not~span~R3|+GFjnC'-F?6#%jnIf~T:~V~-->~W~is~a~l inear~transformation~and~ker(T)=|fr0|hr,~then~|+G-F?6#%?(a)~T~is~a~one-to-one~m apping|+G-F?6#%H(b)~T~is~a~one-to-one~and~onto~mapping|+G-F?6#%C(c)~T~is~not~a~ one-to-one~mapping|+G-F?6#%L(d)~T~is~not~a~one-to-one~mapping~but~onto|+GF]qC'- F?6#%inIf~T:~R3~-->~R2~is~defined~by~T(x1,x2,x3)~=~(x1-x2,x3),~then~|+G-F?6#%E( a)~a~basis~for~ker(T)~is~|fr(0,0,0)|hr|+G-F?6#%F(b)~a~basis~for~ker(T)~is~|fr(1 ,1,0)|hr~|+G-F?6#%N(c)~a~basis~for~ker(T)~is~|fr(1,1,0),(0,0,1)|hr|hr|+G-F?6#%E (d)~a~basis~for~ker(T)~is~|fr(0,0,1)|hr|+GF_rC'Fgj_l-F?6#%N(a)~~the~range(T)~is ~a~proper~subspace~of~R2|+G-F?6#%K(b)~~the~range(T)~is~not~a~subspace~of~R3|+G- F?6#%9(c)~~the~range(T)~is~R2|+G-F?6#%9(d)~~the~range(T)~is~R3|+GFasC'-F?6#%TA~ 5x4~matrix~A~represents~a~linear~transformation~|+G-F?6#%4(a)~from~R5~to~R4~|+G -F?6#%:(b)~which~is~one-to-one~|+G-F?6#%4(c)~which~is~onto~|+G-F?6#%3(d)~from~R 4~to~R5~GFctC)-F?6#%hnIf~T:~R3~-->~R2~is~defined~on~the~standard~basis~as~follo ws:|+G-F?6#%en~T(1,0,0)~=~(1,1)~;~T(0,1,0)~=~(2,1)~;~T(0,0,1)~=~(-1,1)~|+G-F?6# %Hthen~the~transformation~T~is~given~by:|+G-F?6#%I(a)~T(x1,x2,x3)~=~(x1+2x2-x3, ~x1+x2+x3)|+G-F?6#%J(b)~T(x1,x2,x3)~=~(x1+2x2-x3,~-x1+x2+x3)|+G-F?6#%I(c)~T(x1, x2,x3)~=~(x1+2x2-x3,~x1+x2-x3)|+G-F?6#%I(d)~T(x1,x2,x3)~=~(x1+2x2+x3,~x1+x2+x3) |+GF_uC*-F?6#%jnIf~a~basis~for~ker(T)~is~|fr(1,1,1)|hr~and~a~basis~for~range(T) ~is|+G-F?6#%[o|fr(1,0,1),~(-1,1,0)|hr~for~T:~R3~-->~R3~,~then~the~solution~to~t he|+G-F?6#%[o~nonhomogenous~system~of~equations~Ax=b~with~A~being~the~matrix|+G -F?6#%8~representation~of~T~:|+G-F?6#%in(a)~is~k1*(1,1,1)~+~span(range(T))~and~ k1~is~any~real~number)|+G-F?6#%V(b)~is~k1*(1,1,1)~and~k1~is~an~arbitrary~real~n umber|+G-F?6#%@(c)~consists~of~span(range(T)~|+G-F?6#%X(d)~is~k1*(1,1,1)~+~(0,1 ,1)~and~k1~is~any~real~number~|+GF[vC+-F?6#%jn~If~T:~R2~-->~R2~is~defined~by~th e~matrix~multiplication~where|+G-F?6#%;~the~matrix~A~is~given~by|+G>Fdo-Fio6#7$ F_fmF_elFgil-F?6#%jnThen~the~inverse~of~transformation~T~is~the~transformation~ T1:|+G-F?6#%5(a)~T1:=~(x1-x2,x2)|+G-F?6#%5(b)~T1:=~(x1,x1-x2)|+G-F?6#%6(c)~T1:= ~(-x1+x2,x2)|+G-F?6#%6(d)~T1:=~(x2,-x1+x2)|+GF^wC*-F?6#%8Given~the~3x4~matrix~A |+G>Fdo-Fio6#7%F\fo7&F~W~is~linear~because~|+G-F?6#%;(a)~it~preserves~additio n|+G-F?6#%T(b)~it~preseves~addition~and~scalar~multiplication|+G-F?6#%H(c)~it~p reserves~scalar~multiplication|+G-F?6#%/(d)~T(0)~=~0~|+GF_alC'-F?6#%MIf~T:~R2~- ->~R2~is~T(x1,x2)~=(x2,x1),~then~|+G-F?6#%F(a)~T~is~a~reflection~on~the~x-axis~ |+G-F?6#%F(b)~T~is~a~reflection~on~the~y-axis~|+G-F?6#%H(c)~T~is~a~reflection~o n~the~line~y=x~|+G-F?6#%H(d)~T~is~a~rotation~around~the~origin~|+GFablC'-F?6#%\ oIf~A~is~a~5x6~matrix,~then~Range(T)~where~T~is~represented~by~A~|+GF``pFc`pFfc nFicnF\^oC'-F?6#%NIf~T:~R2~-->~R2~is~T(x1,x2)~=(-x1,x2),~then~|+G-F?6#%F(a)~T~i s~a~reflection~on~the~y-axis~|+G-F?6#%F(b)~T~is~a~reflection~on~the~x-axis~|+GF ig`lF\h`lFh^oC'-F?6#%OIf~T:~R2~-->~R2~is~T(x1,x2)~=(-x1,-x2),~then~|+GFcg`lFfg` lFig`l-F?6#%J(d)~T~is~a~reflection~around~the~origin~|+GFa`xC'-F?6#%NIf~T:~R2~- ->~R2~is~T(x1,x2)~=(x1,-x2),~then~|+GFcg`lFfg`lFig`lF\h`lFj`xC(-F?6#%inIf~T1:~R 2~-->~R2~is~T1(x1,x2)~=(-x1,x2)~and~T2:~R2~-->~R2~is~|+G-F?6#%N~T2(x1,x2)~=(-x1 ,-x2),~then~the~composition~|+G-F?6#%N(a)~(T1~o~T2)~is~a~reflection~on~the~x-ax is~|+G-F?6#%N(b)~(T1~o~T2)~is~a~reflection~on~the~y-axis~|+G-F?6#%P(c)~(T1~o~T2 )~is~a~reflection~on~the~line~y=x~|+G-F?6#%P(d)~(T1~o~T2)~is~a~rotation~around~ the~origin~|+GF`axC(-F?6#%gnIf~T1:~R2~->~R2~is~T1(x1,x2)~=(-x1,x2)~and~T2:~R2~- >~R2~is~|+G-F?6#%L~T2(x1,x2)~=(x1,-x2),~then~the~composition|+GF_j`lFbj`lFej`l- F?6#%Q(d)~(T1~o~T2)~is~a~reflection~about~the~origin~|+G/FLFge]lC'-F?6#%[oA~one -to-one~and~onto~linear~transformation~on~a~linear~space~V|+G-F?6#%7(a)~is~not~ invertible|+G-F?6#%3(b)~is~invertible|+G-F?6#%I(c)~is~represented~by~a~singular ~matrix|+G-F?6#%L(d)~is~represented~by~an~orthogonal~matrix|+G/FLFf^_lC'-F?6#%g oThe~kernel~of~one-to-one~and~onto~linear~transformation~of~a~linear~space~V|+G -F?6#%5(a)~is~an~empty~set|+G-F?6#%7(b)~is~of~dimension~1|+G-F?6#%A(c)~consists ~of~the~zero~vector|+G-F?6#%D(d)~is~equal~to~the~linear~space~V|+G/FLFg^_lC(-F? 6#%[oThe~kernel~of~a~linear~transformation~T:~V-->W~for~which~T(v)=0|+G-F?6#%0f or~all~v~in~V|+GF[]al-F?6#%D(b)~is~a~proper~non-empty~set~of~V|+G-F?6#%E(c)~is~ equal~to~the~linear~space~V~|+G-F?6#%A(d)~consists~of~the~zero~vector|+G/FLFh^_ lC(-F?6#%jnThe~range~of~a~linear~transformation~T:~V-->W~for~which~T(v)=0|+GF\^ alF[]al-F?6#%I(b)~is~a~proper~non-trivial~subset~of~W|+G-F?6#%E(c)~is~equal~to~ the~linear~space~W~|+G-F?6#%8(d)~is~the~zero~vector|+G/FLFi^_lC(-F?6#%jnThe~ran ge~of~a~linear~transformation~T:~V-->V~for~which~T(v)=v|+GF\^alF[]al-F?6#%8(b)~ is~a~subspace~of~V|+GFb^alFc_al/FLFj^_lC)-F?6#%jnLet~T:R3-->R4~be~a~linear~tran sformation~whose~kernel~consists|+G-F?6#%inof~the~set~of~all~vectors~|fr(x1,x2, x3)~|gr~x1-x2+x3=0|hr.~Then~the|+G-F?6#%@dimension~of~the~range~of~T~is|+G-F?6# %'(a)~1|+GF_]o-F?6#%'(c)~3|+GF^_o/FLF[__lC)F``al-F?6#%fnof~the~set~of~all~vecto rs~|fr(x1,x2,x3)~|gr~x2=x3=0|hr.~Then~the|+GFf`alFi`alF_]oF\aalF^_o/FLF\__lC(F` `al-F?6#%\oof~the~zero~vector~only.~Then~the~dimension~of~the~range~of~T~is|+GF i`alF_]oF\aalF^_o/FLF]__lC(-F?6#%jnLet~T:R4-->R4~be~a~linear~transformation~who se~kernel~consists|+GFfaalFi`alF_]oF\aalF^_o/FLF^__lC(-F?6#%gnLet~T:R4-->R4~be~ a~linear~transformation~whose~kernel~is~R4|+G-F?6#%IThen~the~dimension~of~the~r ange~of~T~is|+G-F?6#%'(a)~0|+GFf[pF`^n-F?6#%'(d)~3|+G/FLFg]_lC)-F?6#%JThe~trans formation~defined~by~the~matrix|+G>Fdo-Fio6#7$F_el7$FFdo-F io6#7$FfelF_elFgilFfcalFicalF\dalF_dal/FLFi]_lC)F^cal>Fdo-Fio6#7$7$FgvFR3~given~by~T(x,y,z) =(x,y,0)~is~an~example|+G-F?6#%E(a)~of~projection~onto~the~xz-plane|+G-F?6#%E(b )~of~projection~onto~the~yz-plane|+G-F?6#%E(c)~of~projection~onto~the~xy-plane| +G-F?6#%H(d)~of~projection~onto~the~plane~x+y=0|+G/FLF[^_lC'-F?6#%jnThe~mapping ~T:~R3-->R3~given~by~T(x,y,z)=(0,y,z)~is~an~example|+GFdealFgealFjeal-F?6#%H(d) ~of~projection~onto~the~plane~x+z=0|+G/FLF\^_lC'-F?6#%jnThe~mapping~T:~R3-->R3~ given~by~T(x,y,z)=(x,0,z)~is~an~example|+GFdealFgealFjealFefal/FLFd__lC,-F?6#%S The~mapping~T:~R3-->R3~given~by~T(x,y,z)=(x,0,z)~|+G-F?6#%Hwith~respect~to~the~ standard~basis~has|+G-F?6#%?(a)~the~matrix~representation|+G-Fao6#/Fdo-Ffo6#-Fi o6#7%Fg`m7%FR3~given~by~T(x,y,z)=(x,z,y)~|+GFbgalFegalFhgalFahal -Fao6#/Fdo-Ffo6#-Fio6#7%Fg`mFicvF[amFghal-Fao6#/FdoF\a\lFbial-Fao6#/Fdo-Ffo6#-F io6#7%Fg`mF`hal7%FR3~given~by~T(x,y,z )=(x,x+y+z,y)~|+GFbgalFegal-Fao6#/Fdo-Ffo6#-Fio6#7%Fg`mF[amFje\lFahal-Fao6#/Fdo -Ffo6#-Fio6#7%Fg`mFicvFje\lFghal-Fao6#/Fdo-Ffo6#-Fio6#7%Fg`mF__mFe[blFbial-Fao6 #/Fdo-Ffo6#-Fio6#7%Fg`mFje\lF[am/FLFg__lC,-F?6#%TThe~mapping~T:~R3-->R3~given~b y~T(x,y,z)=(x,0,-z)~|+GFbgalFegal-Fao6#/Fdo-Ffo6#-Fio6#7%Fg`mF[amFe]nFahal-Fao6 #/Fdo-Ffo6#-Fio6#7%Fg`mF`halF``mFghal-Fao6#/Fdo-Ffo6#-Fio6#7%Fg`mF`halFe]nFbial -Fao6#/Fdo-Ffo6#-Fio6#7%Fg`mF`hal7%FFFFFgv/FLF___lC(-F?6#%YLet~T:R5-->R3~be~a~l inear~transformation~with~nullity~3|+GFcbalFfbalFf[pF`^nFibal/FLF`__lC(-F?6#%YL et~T:R5-->R4~be~a~linear~transformation~with~nullity~1|+G-F?6#%9Then~the~range~ of~T~is~|+G-F?6#%E(a)~a~subspace~of~R4~of~dimension~1|+G-F?6#%E(b)~a~subspace~o f~R4~of~dimension~3|+G-F?6#%E(c)~a~subspace~of~R4~of~dimension~2|+G-F?6#%E(d)~a ~subspace~of~R4~of~dimension~4|+G/FLFh__lC0-F?6#%inIf~every~point~(x,y)~in~R2~i s~rotated~counter~clockwise~by~45|+G-F?6#%\odegrees~then,~the~matrix~that~repre sents~this~transformation~is:|+G>F[el-Fio6#7$7$*$-%%sqrtGFg]rFgvFgbbl7$,$FgbblF gvFgbbl>Fbel-Fio6#7$FjbblFfbbl>Fhel-Fio6#7$Ffbbl7$F[cblF[cbl>Figl-Fio6#7$FfbblF fbbl-Fao6$%$A1=GF[el-Fao6$%$A2=GFbel-Fao6$%$A3=GFhel-Fao6$%$A4=GFigl-F?6#%((a)~ A1|+G-F?6#%((b)~A2|+G-F?6#%((c)~A3|+G-F?6#%((d)~A4|+G/FLF]^_lC(-F?6#%^oThe~poin ts~that~are~invariant~with~respect~to~the~reflection~given|+G-F?6#%Iby~the~tran sformation:~T(x,y)=(y,x)~are|+G-F?6#%O(a)~the~points~that~belong~to~the~line~y~ =~-x|+G-F?6#%N(b)~the~points~that~belong~to~the~line~y~=~x|+G-F?6#%O(c)~the~poi nts~that~belong~to~the~line~y~=~2x|+G-F?6#%P(d)~the~points~that~belong~to~the~l ine~y~=~-2x|+G/FLF^^_lC(-F?6#%[oThe~point(s)~that~are~invariant~wrt~the~reflect ion~given~by~the|+G-F?6#%Dtransformation:~T(x,y)=(-x,-y)~are|+GFieblF\fbl-F?6#% 6(c)~the~origin~(0,0)|+G-F?6#%L(d)~the~origin~(0,0)~and~the~point~(-1,-1)|+G/FL F_^_lC(-F?6#%jnThe~point(s)~that~are~invariant~with~respect~to~the~reflection|+ G-F?6#%Pgiven~by~the~transformation:~T(x,y)=(-x,y)~are|+G-F?6#%J(a)~the~points~ that~belong~to~the~x-axis|+G-F?6#%J(b)~the~points~that~belong~to~the~y-axis|+GF ]gbl-F?6#%K(d)~the~origin~(0,0)~and~the~point~(-1,1)|+G/FLF`^_lC(Fegbl-F?6#%Pgi ven~by~the~transformation:~T(x,y)=(x,-y)~are|+GF[hblF^hblF]gblFahbl/FLFa^_lC)F^ cal>Fdo-Fio6#7$FcfmF\[\l-Fao6$%$A:=GFdoFfcalFicalF\dalFcx/FLFb^_lC'-F?6#%ZThe~t ransformation~T:~R2~-->~R2~given~by~T(x,y)=(xy,x+y)|+G-F?6#%H(a)~is~not~linear~ because~T(0,0)=(0,0)|+G-F?6#%Y(b)~is~not~linear~because~it~does~not~preserve~ad dition|+G-F?6#%D(c)~is~linear~because~T(1,1)=(1,2)|+G-F?6#%D(d)~is~linear~becau se~T(0,0)=(0,0)|+G/FLFc^_lC'-F?6#%enThe~transformation~T:~R2~-->~R2~given~by~T( x,y)=(x+2,y+1)|+G-F?6#%H(a)~is~not~linear~because~T(0,0)=(2,1)|+G-F?6#%S(b)~is~ linear~because~it~represents~a~translation|+G-F?6#%D(c)~is~linear~because~T(1,1 )=(3,2)|+G-F?6#%fn(d)~is~not~linear~because~its~nullity~is~not~equal~to~zero|+G /FLFi__lC'-F?6#%ZThe~transformation~T:~R2~-->~R2~given~by~T(x,y)=(-x,-y)~|+G-F? 6#%D(a)~is~not~a~linear~transformation|+G-F?6#%L(b)~is~an~invertible~linear~tra nsformation|+G-F?6#%>(c)~has~a~non~trivial~kernel|+G-F?6#%I(d)~is~not~an~invert ible~transformation|+G/FLF^`_lC(-F?6#%hnThe~components~of~the~vector~[1,2]~with ~respect~to~the~basis|+G-F?6#%2|fr[1,1],[1,0]|hr~is|+G-F?6#%+(a)~[2,1]|+G-F?6#% ,(b)~[-2,1]|+G-F?6#%,(c)~[2,-1]|+G-F?6#%-(d)~[-2,-1]|+G/FLFj__lC(-F?6#%jnThe~tr ansition~matrix~that~changes~the~coordinates~of~a~vector|+G-F?6#%hnfrom~the~sta ndard~basis~to~a~basis~|fr[1,1],[2,1]|hr~is~given~by|+G-F?6#%U(a)~is~the~matrix ~whose~columns~are~|fr[1,-2],[-2,1]|hr|+G-F?6#%U(b)~is~the~matrix~whose~columns ~are~|fr[-1,1],[2,-1]|hr|+G-F?6#%T(c)~is~the~matrix~whose~columns~are~|fr[1,-2] ,[2,1]|hr|+G-F?6#%S(d)~is~the~matrix~whose~columns~are~|fr[1,2],[2,1]|hr|+G/FLF _`_lC)-F?6#%^oIf~S1~is~the~transition~matrix~from~basis~B1~to~standard~basis~an d|+G-F?6#%[oS2~is~the~transition~matrix~from~basis~B2~to~the~standard~basis|+G- F?6#%Mthen~the~transition~matrix~from~B1~to~B2~is|+G-F?6#%4(a)~inverse(S1)*S2|+ G-F?6#%4(b)~inverse(S2)*S1|+G-F?6#%+(c)~S1*S2|+G-F?6#%4(d)~inverse(S1*S2)|+G/FL F``_lC)F__clFb_cl-F?6#%Mthen~the~transition~matrix~from~B2~to~B1~is|+GFh_clF[`c lF^`clFa`cl/FLFa`_lC)-F?6#%TConsider~a~transformation~given~by~T([x,y])=[-x,y]| +G-F?6#%ZThe~matrix~representation~of~T~with~respect~to~the~basis|+GFj\cl-F?6#% U(a)~The~matrix~whose~rows~are~given~by~[-1,0],[0,1]|+G-F?6#%U(b)~The~matrix~wh ose~rows~are~given~by~[1,0],[0,-1]|+G-F?6#%Y(c)~The~matrix~whose~columns~are~gi ven~by~[0,1],[-1,-1]|+G-F?6#%X(d)~The~matrix~whose~columns~are~given~by~[0,1],[ -1,1]|+G/FLFb`_lC)F[aclF^acl-F?6#%3|fr[0,1],[1,-1]|hr~is|+G-F?6#%V(a)~The~matri x~whose~rows~are~given~by~[-1,1],[2,-1]|+G-F?6#%V(b)~The~matrix~whose~rows~are~ given~by~[-1,1],[-1,2]|+G-F?6#%W(c)~The~matrix~whose~columns~are~given~by~[0,1] ,[1,2]|+G-F?6#%Y(d)~The~matrix~whose~columns~are~given~by~[1,0],[-2,-1]|+G/FLF[ `_lC(-F?6#%\oThe~matrix~representation~of~T([x,y,z])=[x,y,0])~with~respect~to|+ G-F?6#%7the~standard~basis~is|+G-F?6#%W(a)~The~matrix~whose~rows~are~[1,0,0],[0 ,1,0],[0,0,0]|+G-F?6#%Z(b)~The~matrix~whose~columns~are~[1,0,0],[0,1,0],[0,0,1] |+G-F?6#%W(c)~The~matrix~whose~rows~are~[1,1,0],[0,1,0],[0,0,0]|+G-F?6#%Z(d)~Th e~matrix~whose~columns~are~[1,1,0],[0,1,0],[0,0,0]|+G/FLFc`_lC(-F?6#%jnThe~coor dinates~of~the~vector~[-1,2]~with~respect~to~the~basis|+GFj\cl-F?6#%,(a)~[2,-1] |+G-F?6#%,(b)~[-2,3]|+G-F?6#%+(c)~[2,3]|+G-F?6#%,(d)~[2,-3]|+G/FLFijqC(-F?6#%[o The~matrix~representation~of~T([x,y,z])=[x+y,z]~with~respect~to|+G-F?6#%+any~ba sis|+G-F?6#%5(a)~is~a~2x3~matrix|+G-F?6#%5(b)~is~a~3x2~matrix|+G-F?6#%5(c)~is~a ~3x3~matrix|+G-F?6#%5(d)~is~a~2x2~matrix|+G/FLFd`_lC(-F?6#%[oConsider~a~transfo rmation~given~by~T([x,y])=[-x,y].~The~matrix~|+G-F?6#%jnrepresentation~of~T~wit h~respect~to~the~basis~|fr[1,0],[0,1]|hr~is|+GFaacl-F?6#%V(b)~The~matrix~whose~ rows~are~given~by~[-1,0],[0,-1]|+G-F?6#%X(c)~The~matrix~whose~columns~are~given ~by~[0,1],[-1,0]|+GFjacl/FLFe`_lC(-F?6#%jnConsider~a~transformation~given~by~T( [x,y])=[-x,y].~The~matrix|+G-F?6#%jnrepresentation~of~T~with~respect~to~the~bas is~|fr[1,1],[1,0]|hr~is|+GFaacl-F?6#%V(b)~The~matrix~whose~rows~are~given~by~[1 ,-2],[0,-1]|+G-F?6#%Y(c)~The~matrix~whose~columns~are~given~by~[1,-2],[0,-1]|+G -F?6#%X(d)~The~matrix~whose~columns~are~given~by~[-1,0],[0,1]|+G/FLFf`_lC(-F?6# %hnConsider~a~transformation~given~by~T([x,y])=[x,y].The~matrix|+GF\gclFaacl-F? 6#%V(b)~The~matrix~whose~rows~are~given~by~[0,1],[-1,-1]|+G-F?6#%W(c)~The~matri x~whose~columns~are~given~by~[1,0],[0,1]|+G-F?6#%X(d)~The~matrix~whose~columns~ are~given~by[0,1],[-1,-1]|+G/FLFg`_lC(-F?6#%\oConsider~a~transformation~given~b y~T([x,y])=[-x,-y].~The~matrix~|+GF\gclFaaclF[icl-F?6#%Y(c)~The~matrix~whose~co lumns~are~given~by~[-1,0],[0,-1]|+GFjacl/FLFh`_lC(-F?6#%[oThe~matrix~representa tion~of~the~linear~transformation~T:R2->R3|+G-F?6#%Gwith~respect~to~the~standar d~basis~is|+G-F?6#%2(a)~a~2x3~matrix|+G-F?6#%2(b)~a~3x2~matrix|+G-F?6#%=(c)~the ~2x2~identity~matrix|+G-F?6#%=(d)~the~3x3~identity~matrix|+G/FLFi`_lC'-F?6#%hnI f~A~and~B~are~similar~matrices~and~A~is~invertible,~then~B~|+G-F?6#%1(a)~is~sin gular|+GF]\al-F?6#%2(c)~is~symmetric|+G-F?6#%6(d)~is~not~symmetric|+G/FLF\a_lC( -F?6#%aoThe~matrix~representation~of~the~linear~transformation~T([x,y])=[y,x]|+ GFajcl-F?6#%9(a)~the~identity~matrix|+G-F?6#%?(b)~the~matrix([[1,0],[1,1]])|+G- F?6#%?(c)~the~matrix([[0,1],[1,0]])|+G-F?6#%?(d)~the~matrix([[1,1],[0,1]])|+G/F LF]a_lC(-F?6#%_oIf~S~and~T~are~the~matrix~representation~of~a~linear~transforma tion|+G-F?6#%Kwith~respect~to~the~bases~B1~and~B2,~then|+G-F?6#%6(a)~S~is~inver tible~|+G-F?6#%5(b)~T~is~invertible|+G-F?6#%7(c)~S~is~similar~to~T|+G-F?6#%=(d) ~S~is~the~transpose~of~T|+G/FLF^a_lC'-F?6#%OIf~a~matrix~A~is~similar~to~a~matri x~B,~then~|+G-F?6#%Y(a)~Both~A~and~B~have~the~same~reduced~row~echelon~form|+G- F?6#%co(b)~inverse~of~A~is~similar~to~inverse~of~B,~if~A~and~B~are~nonsingular| +G-F?6#%?(c)~AB~is~the~identity~matrix|+G-F?6#%Y(d)~Either~transpose(A)~=~B~OR~ transpose(B)=A~must~hold|+G/FLFdjqC'-F?6#%GIf~A~and~B~are~similar~matrices,~the n|+G-F?6#%Z(a)~A~and~B~must~have~the~same~row~and~column~dimensions|+G-F?6#%D(b )~AB~must~be~the~identity~matrix|+G-F?6#%W(c)~A~and~B~are~row~equivalent~to~the ~identity~matrix|+G-F?6#%D(d)~BA~must~be~the~identity~matrix|+G/FLF_a_lC(-F?6#% OFind~a~matrix~similar~to~the~following~matrix|+G-Fao6#-Fio6#7%Fg`mFijzFe]n-F?6 #%G(a)~matrix([[1,0,0],[0,1,1],[1,1,1]])|+G-F?6#%I(b)~matrix([[-1,0,0],[0,-1,0] ,[0,0,1]])|+G-F?6#%I(c)~matrix([[1,0,0],[0,-1,0],[0,0,-1]])|+GFcx/FLF`a_lC)-F?6 #%^oIf~A~is~the~matrix~represenation~of~a~linear~transformation~T~with|+G-F?6#% ^orespect~to~a~basis~B1~and~B~is~the~matrix~representation~of~T~with|+G-F?6#%=r espect~to~a~basis~B2,~then|+G-F?6#%G(a)~A~is~equal~to~the~transpose~of~B~|+G-F? 6#%7(b)~A~is~similar~to~B|+G-F?6#%<(c)~A~and~B~are~invertible|+G-F?6#%;(d)~B~is ~the~inverse~of~A|+G/FLFaa_lC'-F?6#%PAny~non-singular~matrix~A~is~similar~to~it self|+G-F?6#%*(a)~True|+G-F?6#%+(b)~False|+G-F?6#%C(c)~Only~if~A~is~upper~trian gular|+G-F?6#%<(d)~Only~if~A~is~symmetric|+G/FLFba_lC'-F?6#%NIf~a~matrix~A~is~s imilar~to~a~matrix~B,~then|+G-F?6#%;(a)~A^n~is~simialr~to~B^n|+G-F?6#%@(b)~B~mu st~be~the~inverse~of~A|+G-F?6#%B(c)~B~must~be~the~transpose~of~A|+G-F?6#%;(d)~A ^T~is~similar~to~B^T|+G/FLFca_lC'-F?6#%HA~matrix~A~is~similar~to~a~matrix~B~if| +G-F?6#%<(a)~B~is~the~inverse~of~A~|+G-F?6#%7(b)~B~is~a~power~of~A|+G-F?6#%T(c) ~A~is~equal~to~inverse(C)*B*C~for~some~matrix~C|+G-F?6#%=(d)~B~is~the~transpose ~of~A|+G-Fao6#%SPlease~add~more~challenging~questions~to~this~set|+GFfhtF7F7F7, I,question2_1:FectFgctF7F7C)-F\htF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Feeq Fcft>FdoFeftF7FfftF7, I/questionset1_1:F$6'F'F/F0F1F2F7F7C&@$F:F>Fecl@AFKC)-F?6#%@The~system~of~linea r~equations|+G-F?6#%.~2x~+~2y~=~5|+G-F?6#%.~~x~+~~y~=~2|+G-F?6#%;(a)~has~a~uniq ue~solution|+G-F?6#%5(b)~has~no~solution|+G-F?6#%C(c)~has~infinitely~many~solut ions|+G-F?6#%6(d)~has~two~solutionsGFjnC'-F?6#%MAny~homogeneous~system~of~linea r~equations:|+G-F?6#%?(a)~has~at~least~one~solution|+G-F?6#%?(b)~has~an~empty~s olution~set|+G-F?6#%C(c)~has~only~the~trivial~solution|+G-F?6#%C(d)~has~only~no ntrivial~solutions|+GF]qC'-F?6#%PAny~nonhomogeneous~system~of~linear~equations: |+G-F?6#%B(a)~has~a~non-empty~solution~set|+GF]jdlF`jdlFcxF_rC*-F?6#%CThe~solut ion~to~the~linear~system|+G-F?6#%.~2x~+~2y~=~0|+G-F?6#%.~~x~-~~y~=~0|+G-F?6#%$i s|+G-F?6#%,(a)~[2,-2]|+G-F?6#%+(b)~[0,0]|+G-F?6#%,(c)~[-1,1]|+GFcxFasC*-F?6#%,T he~system|+G-F?6#%3~2x~-~2y~-~2z~=~0|+G-F?6#%3-2x~+~2y~+~~z~=~1|+G-F?6#%3~~x~-~ ~y~-~~z~=~2|+G-F?6#%4(a)~has~a~solution|+G-F?6#%6(b)~is~a~homogeneous|+G-F?6#%> (c)~does~not~have~a~solution|+G-F?6#%;(d)~has~a~unique~solution|+GFctC'-F?6#%UT he~graphs~of~x~-~6y~+~z~=~0~and~2x~-~12y~+~2z~=~1~|+G-F?6#%:(a)~intersect~at~a~ point|+G-F?6#%9(b)~intersect~at~a~line|+G-F?6#%9(c)~parallel~planes~~~~|+G-F?6# %8(d)~overlapping~planes|+GF_uC)-F?6#%_oThe~system~of~linear~equations~x+2y-z=0 ~and~x-y+z=1~has~a~solution.|+G-F?6#%^oSuppose~we~add~the~new~equation~3x+3y-z= k~to~these~equations.~Then|+G-F?6#%9the~resulting~system~is|+G-F?6#%V(a)~has~in finitely~many~solutions~for~any~value~of~k|+G-F?6#%N(b)~has~a~unique~solution~f or~any~value~of~k|+G-F?6#%fn(c)~has~an~empty~solution~set~regardless~of~the~val ue~of~k|+G-F?6#%J(d)~has~infinitely~many~solutions~if~k=1|+GF[vC)-F?6#%_oIf~the ~solution~set~of~a~non-homogeneous~system~of~linear~equations|+G-F?6#%]ois~give n~by~x0+tx1,~then~a~solution~of~the~associated~homogeneous|+G-F?6#%,system~is:| +G-F?6#%+(a)~x0-x1|+G-F?6#%((b)~x0|+G-F?6#%((c)~x1|+G-F?6#%+(d)~x0+x1|+GF^wC'-F ?6#%IThe~two~equations~x+2y-z=0~and~x-y+z=1~|+G-F?6#%T(a)~intersect~at~a~line~p assing~through~the~origin|+G-F?6#%en(b)~intersect~at~a~line~passing~through~the ~point~(0,1,2)|+G-F?6#%A(c)~intersect~at~a~single~point|+G-F?6#%en(d)~intersect ~at~a~line~passing~through~the~point~(1,1,0)|+GFdwC'-F?6#%KThe~two~equations~2x +2y-2z=6~and~x+y-z=1~|+GF\bel-F?6#%en(b)~intersect~at~a~line~passing~through~th e~point~(0,2,1)|+G-F?6#%D(c)~represents~two~parallel~planes|+G-F?6#%fn(d)~inter sect~at~a~line~passing~through~the~point~(0,1,-2)|+GFfxC(-F?6#%_oIf~three~plane s~have~a~line~in~common,~then~the~system~of~equations|+G-F?6#%=represented~by~t hese~planes|+GFjhdl-F?6#%C(b)~has~infinitely~many~solutions|+G-F?6#%5(c)~has~no ~solution|+GFcxF\[lC(-F?6#%YIf~three~planes~have~a~point~in~common,~then~the~sy stem|+G-F?6#%Mof~linear~equations~defined~by~these~planes|+G-F?6#%C(a)~has~infi nitely~many~solutions|+G-F?6#%;(b)~has~a~unique~solution|+GF_del-F?6#%G(d)~has~ only~three~distinct~solutions|+GFa_lC(-F?6#%_oIf~three~planes~intersect~such~th at~any~two~have~a~distinct~line~in|+G-F?6#%^ocommon,~the~the~system~of~linear~e quations~defined~by~these~planes|+G-F?6#%A(a)~must~have~a~unique~solution|+G-F? 6#%I(b)~must~have~infinitely~many~solutions|+G-F?6#%:(c)~must~be~inconsistent|+ G-F?6#%H(d)~none~of~the~above~will~always~hold|+GF_alC(-F?6#%]oIf~three~lines~i ntersect~such~that~any~two~have~distinct~point~in|+G-F?6#%^ocommon,~then~the~sy stem~of~linear~equations~defined~by~these~lines|+GFieelF\felF_felFbfelFablC'-F? 6#%aoWhich~one~of~the~following~statements~is~correct?~The~solution~set~of|+G-F ?6#%\o(a)~every~nonhomogeneous~system~of~linear~equations~is~non-empty|+G-F?6#% in(b)~every~homogeneous~system~of~linear~equations~is~non-empty|+G-F?6#%en(c)~e very~homogeneous~system~of~linear~equations~is~empty|+G-F?6#%hn(d)~Every~nonhom ogeneous~system~of~linear~equations~is~empty|+G-Fao6#%@Add~More~Questions~to~th e~test!G-F?6#%1|+**************|+GF7F7F7, I,question2_2:FectFgctF7F7C)-FbavF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Feeq Fcft>FdoFeftF7FfftF7, I/questionset1_2:F$F\hdlF7F7C&@$F:C$F>FCFecl@AFKC'Fgidl-F?6#%:(a)~is~always~con sistent|+G-F?6#%<(b)~is~always~inconsistent|+G-F?6#%I(c)~does~only~have~the~tri vial~solution|+G-F?6#%I(d)~does~only~have~nontrivial~solutions|+GFjnC'FgjdlF^je lFajelFdjelFcxF]qC)-F?6#%]oThe~system~of~linear~equations~x+2y-z=0~and~x-y+z=1~ is~consistent|+G-F?6#%inSuppose~we~add~the~new~equation~3x+3y-z=k~to~these~equa tions.|+G-F?6#%;Then~the~resulting~system|+G-F?6#%_o(a)~is~consistent~with~infi nitely~many~solutions~for~any~value~of~k|+G-F?6#%gn(b)~is~consistent~with~a~uni que~solution~for~any~value~of~k|+G-F?6#%Y(c)~is~always~inconsistent~regardless~ of~the~value~of~k|+G-F?6#%Y(d)~is~consistent~with~infinitely~many~solutions~if~ k=1|+GF_rC)-F?6#%^oIf~the~solution~set~of~a~nonhomogeneous~system~of~linear~equ ations|+G-F?6#%^ois~given~by~x0+t.x1,~then~a~solution~of~the~associated~homogen eous|+GFi`elF\aelF_aelFbaelFeaelFasC)-F?6#%\oA~nonhomogeneous~system~of~linear~ equations~has~infinitely~many~|+G-F?6#%\osolutions~given~by~x~=~(1,2,-4)~+~t*(- 1,4,5)~where~t~is~any~real|+G-F?6#%\oparameter.Then~the~solution~of~the~associa ted~homogeneous~system|+G-F?6#%4(a)~is~t*(-1,4,5)~|+G-F?6#%<(b)~is~(1-t,2+4*t,- 4-5*t)~|+G-F?6#%B(c)~is~only~the~trivial~solution|+G-F?6#%1(d)~is~(0,6,1)~|+GFc tC(-F?6#%]oA~nonhomogeneous~system~of~linear~equations~has~a~unique~solution|+G -F?6#%hngiven~by~x=(-1,0,3,5).Then~the~associated~homogeneous~system|+G-F?6#%G( a)~has~t*(-1,0,3,5)~as~its~solution~|+GF]idlF`jdl-F?6#%O(d)~has~a~solution~that ~cannot~be~determined~|+GF_uC)-F?6#%]oThe~system~of~linear~equations~x+2y-z=1~a nd~x-y+z=0~is~consistent|+GF_[flFb[fl-F?6#%L(a)~is~inconsistent~if~k~is~not~equ al~to~2|+GFh[fl-F?6#%_o(c)~is~consistent~with~infinitely~many~solutions~for~any ~value~of~k|+G-F?6#%H(d)~is~inconsistent~if~k~is~equal~to~2|+GF[vC(-F?6#%@~~The ~system~~2x~-~2y~-~2z~=~4|+G-F?6#%@~~~~~~~~~~~~~~~x~-~~y~-~~z~=~2|+G-F?6#%5(a)~ is~inconsistent|+GFc]el-F?6#%3(c)~is~consistent|+GFi]elF^wC'F]io-F?6#%]o(a)~Eve ry~nonhomogeneous~system~of~linear~equations~is~consistent|+G-F?6#%jn(b)~Every~ homogeneous~system~of~linear~equations~is~consistent|+G-F?6#%\o(c)~Every~homoge neous~system~of~linear~equations~is~inconsistent|+G-F?6#%_o(d)~Every~nonhomogen eous~system~of~linear~equations~is~inconsistent|+GFdwC*Fd\el-F?6#%6~~~x~-~3*y~- ~k*z~=~0|+G-F?6#%6-2*x~+~2*y~+~~~z~=~1|+G-F?6#%6~~-x~-~~y~-~~~~z~=~2|+G-F?6#%E( a)~is~consistent~for~k~equals~to~2|+G-F?6#%H(b)~is~consistent~for~k~not~equal~t o~2|+G-F?6#%Q(c)~is~consistent~regardless~of~the~values~of~k|+GFc]zFfxC(Fh\x-F? 6#%VEvery~nonhomogeneous~system~of~linear~equations~with|+G-F?6#%V(a)~more~equa tions~than~unknowns~may~be~inconsistent|+G-F?6#%jn(b)~less~equations~than~unkno wns~has~infinitely~many~solutions|+G-F?6#%X(c)~less~equations~than~unknowns~has ~a~unique~solution|+G-F?6#%T(d)~more~equations~than~unknowns~may~be~consistent| +GF\[lC(-F?6#%_oIf~a~homogeneous~system~of~n~equations~with~n~unknowns~has~only ~the|+G-F?6#%gozero~solution,~then~the~solution~of~the~associated~nonhomogeneou s~system~is|+G-F?6#%1(a)~is~infinite|+G-F?6#%4(b)~is~a~singleton|+G-F?6#%.(c)~i s~empty|+G-F?6#%2(d)~is~undefined|+GFa_lC(-F?6#%fnIf~a~nonhomogeneous~system~ha s~a~unique~solution,~then~the|+G-F?6#%Vsolution~set~of~the~associated~homogeneo us~system~is|+GF\dfl-F?6#%.(b)~is~empty|+G-F?6#%4(c)~is~a~singleton|+GFedflF_al C(-F?6#%hnIf~a~homogeneous~system~has~more~than~one~solution,~then~the|+G-F?6#% Ysolution~set~of~the~associated~nonhomogeneous~system~is|+GF\dfl-F?6#%/(b)~is~u nique|+GFbdfl-F?6#%:(d)~cannot~be~determined|+GFablC*Fd\el-F?6#%4~~~x~+~y~-~k*z ~=~0|+G-F?6#%4~~~x~-~y~+~~~z~=~1|+G-F?6#%4~~~~~~~(1-k)*z~=~2|+G-F?6#%E(a)~is~co nsistent~for~k~equals~to~1|+G-F?6#%H(b)~is~consistent~for~k~not~equal~to~1|+GFb bflF`^zF\hel-F?6#%7|+********************|+GF7F7F7, I,question2_3:FectFgctF7F7C)-Fe`wF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Feeq Fcft>FdoFeftF7FfftF7, I/questionset1_3:F$F\hdlF7F7C&@$F:F>Fecl@AFKC)-F?6#%-The~matrix~|+G>Fdo-Fio6#7% Fg`mFhcvFicvFgilF_iw-F?6#%I(b)~is~an~example~of~an~identity~matrix|+G-F?6#%=(c) ~is~an~elementary~matrix|+GFcxFjnC*F]\v>Fdo-Fio6#7&7&FFFFdoF_jflFgil-F?6#%inThen~the~matrix~is~not~an~exam ple~of~an~elementary~matrix~if~|+G-F?6#%?(a)~a1~=~0~and~a2~is~not~zero|+GF[[gl- F?6#%7(c)~a1~=~0~and~a2~=~0|+GFcxF_rC,-F?6#%DThe~system~of~linear~equations~A:~ |+G-Fao6#/F\`uFF-Fao6#/,&FevF[oFfvF]wFF-F?6#%@is~equivalent~to~the~system~B:|+G Fc\gl-Fao6#/,&FevFbsFfv!"'F`r-F?6#%in(a)~since~B~is~obtained~from~A~by~an~eleme ntary~row~operation|+G-F?6#%V(b)~since~the~two~systems~have~the~same~solution~s et|+G-F?6#%fn(c)~since~the~two~systems~represent~two~intersecting~lines|+G-F?6# %N(d)~answer(a)~and~answer(b)~are~both~correct|+GFasC,F`\gl-Fao6#/,(FevFFFfvFFF hvFFFF-Fao6#/,(FevFFFfvFgvFhvFFFFFj\glF_^gl-Fao6#/,&FevFFFhvFFFF-F?6#%gn(a)~sin ce~B~is~obtained~from~A~by~elementary~row~operations|+GFe]gl-F?6#%gn(c)~since~t he~two~systems~represent~two~intersecting~planes|+GF[^glFctC,F`\glFc\gl-Fao6#/, &FevF[oFfvF^qFF-F?6#%Jcan~never~be~equivalent~to~the~system~B:|+GFc\gl-Fao6#/Fe _glFbs-F?6#%`o(a)~since~B~cannot~be~obtained~from~A~by~an~elementary~row~operat ion|+G-F?6#%en(b)~since~the~two~systems~have~the~different~solution~set|+GFh]gl F[^glF_uC(-F?6#%]oIf~a~homogeneous~system~of~n~equations~in~n~unknowns~has~only ~the|+GFicflF\dflF_dflFbdflFedflF[vC(-F?6#%`oIf~a~nonhomogeneous~system~of~line ar~equations~has~a~unique~solution|+G-F?6#%inthen~the~solution~set~of~the~assoc iated~homogeneous~system~is|+GF\dflF_eflFbeflFedflF^wC(-F?6#%boIf~a~homogeneous ~system~of~linear~equations~has~more~than~one~solution|+G-F?6#%\othen~the~solut ion~set~of~the~associated~nonhomogeneous~system~is|+GF\dflF\fflFbdflF_fflFdwC*F d\elFcfflFffflFifflF\gflF_gflFbbflF`^zFfxC*Fd\elFcfflFfffl-F?6#%4~~~~~~~(1-k)*z ~=~1|+G-F?6#%N(a)~has~a~unique~solution~if~k~is~equal~to~1|+G-F?6#%R(b)~has~a~u nique~solution~if~k~is~not~equal~to~1|+G-F?6#%Y(c)~has~a~unique~solution~regard less~of~the~values~of~k|+G-F?6#%F(d)~can~never~have~a~unique~solution|+GF\[lC*F d\elFcfflFfffl-F?6#%4~2*x~+~(1-k)*z~=~1|+G-F?6#%V(a)~has~infinitely~many~soluti ons~if~k~is~equal~to~1|+G-F?6#%Z(b)~has~infinitely~many~solutions~if~k~is~not~e qual~to~1|+G-F?6#%[o(c)~has~infinitely~many~solutions~regardless~of~the~values~ of~k|+G-F?6#%N(d)~can~never~have~infinitely~many~solutions|+GFa_lC*Fd\elFcffl-F ?6#%4~~~x~-~y~+~2*z~=~1|+G-F?6#%4~2*x~+~(2-k)*z~=~3|+G-F?6#%W(a)~has~infinitely ~many~solutions~if~k~is~equal~to~-2|+G-F?6#%Z(b)~has~infinitely~many~solutions~ if~k~is~not~equal~to~2|+GF_cglFc]zF_alC*Fd\elFcfflFfcglFicgl-F?6#%<(a)~is~alway s~inconsistent|+G-F?6#%J(b)~is~consistent~if~k~is~not~equal~to~2|+G-F?6#%F(c)~i s~consistent~if~k~is~equal~to~2|+G-F?6#%<(d)~is~inconsistent~if~k=2|+GFablC(-F? 6#%BThe~solution~set~of~the~equation|+G-F?6#%2~~~x~+~y~-~z~=~0|+G-F?6#%@(a)~inc ludes~one~free~variable|+G-F?6#%A(b)~includes~two~free~variables|+G-F?6#%C(c)~i ncludes~three~free~variables|+GFcxF\helFbgflF7F7F7, I,question2_4:FectFgctF7F7C)-FbdxF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Feeq Fcft>FdoFeftF7FfftF7, I/questionset1_4:F$F\hdlF7F7C&@$F:F>Fecl@AFKC0-F?6#%7The~augmented~matrix~|+G>F do-Fio6#7%7%F[oFFFFFhcvF`halFgil-F?6#%<(a)~represents~the~system:|+G-Fao6#/,&Fe vF[oFfvFFFF-Fao6#/FfvF[o-F?6#%<(b)~represents~the~system:|+GFehgl-Fao6#/FcitF[o -F?6#%<(c)~represents~the~system:|+GFehglF_igl-Fao6#/FFdoFb\uFg il-F?6#%SThen~the~system~has~infinitely~many~solutions~if~|+GF`]uFc]u-F?6#%-(c) ~a2=3*a1|+GFi]uF_uC*Fg\hlFfgp>FdoF`bv-FaoF[jl-F?6#%in(a)~The~system~has~infinit ely~many~solutions~for~any~a1,a2,a3|+G-F?6#%X(b)~The~system~has~a~unique~soluti on~for~any~a1,a2,~a3|+GFjbvF]cvF[vC)F\[flF`_el-F?6#%6the~resulting~system|+GFe[ flFh[flF[\flF^\flF^wC+-F?6#%UThe~augmented~matrix~associated~with~a~given~syste m|+G-F?6#%=of~homogeneous~equations~is|+G>Fdo-Fio6#7%7'FFFgvFFFFdo-Fio6#7%7(FFFgvFFFFFFF<7(FFdo-Fio6#7$F``h l7(FFdo-Fio6#7%F``hl7( FFdo-Fio6#7%7'FFgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFhbs>F_ftFaft>Fe eqFcft>FdoFeftF7FfftF7, I,question2_6:FectFgctF7F7C)-FdfzF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Feeq Fcft>FdoFeftF7FfftF7, I/questionset1_5:F$F\hdlF7F7C&@$F:C$F>FCFecl@KFKC)F`ifl>Fdo-Fio6#7%Fe[blFicvF`h alFc]hl-F?6#%=(a)~is~in~row~echelon~form~|+G-F?6#%@(b)~is~in~reduced~echelon~fo rm|+GF\hxFcxFjnC(-F?6#%[oThe~gauss~elimination~algorithm~reduces~the~augmented~ matrix~of|+G-F?6#%Jan~associated~system~of~linear~equations|+G-F?6#%D(a)~to~a~m atrix~in~an~echelon~form|+G-F?6#%C(b)~to~an~upper~triangular~matrix|+G-F?6#%B(c )~to~a~lower~triangular~matrix|+G-F?6#%=(d)~to~a~tridiagonal~matrix|+GF]qC+Fcgp Ffgp>FdoFb\uFc]hlF[]hlF`]uFc]uF^]hlFi]uF_rC*FcgpFfgp>FdoF`bvFc]hlFd]hlFg]hlFjbv F]cvFasC)F\[flF`_elFc_el-F?6#%\o(a)~consistent~with~infinitely~many~solutions~f or~any~value~of~k|+G-F?6#%Z(b)~consistent~with~a~unique~solution~for~any~value~ of~k|+G-F?6#%V(c)~always~inconsistent~regardless~of~the~value~of~k|+G-F?6#%V(d) ~consistent~with~infinitely~many~solutions~if~k=1|+GFctC+-F?6#%\oThe~echelon~fo rm~of~the~augmented~matrix~associated~with~a~given|+G-F?6#%Dsystem~of~homogeneo us~equations~is|+G>FdoFf^hlFgilF\_hlF__hlFb_hlFe_hlFh_hlF_uC+FaihlFdihl>FdoF]`h lFgilF\_hlF__hlFb_hlFe_hlFh_hlF[vC+FaihlFdihl>FdoFe`hlFgilF\_hlF__hlFb_hlFe_hlF h_hlF^wC+FaihlFdihl>FdoF[ahlFgilF\_hlF__hlFb_hlFe_hlFh_hlFdwC)-F?6#%6The~matrix ~given~by:|+G>Fdo-Fio6#7%FicvFe[blF`halFgil-F?6#%<(a)~is~in~row~echelon~form|+G -F?6#%D(b)~is~in~reduced~row~echelon~form|+G-F?6#%@(c)~is~not~in~row~echelon~fo rm|+G-F?6#%jn(d)~is~in~row~echelon~form~but~not~in~reduced~row~echelon~form|+GF fxC'-F?6#%`oThe~backsubstitution~algorithm~should~be~implemented~after~computin g|+G-F?6#%\o(a)~the~inverse~of~the~augmented~matrix~associated~with~a~system|+G -F?6#%^o(b)~the~transpose~of~the~augmented~matrix~associated~with~a~system|+G-F ?6#%`o(c)~the~row~echelon~of~the~augmented~matrix~associated~with~a~system|+G-F ?6#%[o(d)~the~square~of~the~augmented~matrix~associated~with~a~system|+GF\[lC*F ggu>FdoFcfhlFgil-F?6#%>is~an~example~of~a~matrix~in|+G-F?6#%7(a)~row~echelon~fo rm~|+G-F?6#%>(b)~reduced~row~echelon~form|+GF\hxFcxFa_lC*Fggu>FdoF\bhlFgil-F?6# %\ois~the~coefficient~matrix~of~a~homogeneous~system~in~row~echelon|+GFebhlFhbh lF[chlF^chlF_alC.Fggu>Fdo-Fio6#7%7'FF[el-Fcy6#7'F agn,&FcgnFgv%#x5GF_]lFcgn,$F]^ilFgvF]^il>Fbel-Fcy6#7'FFhel -Fcy6#7'F<,$FcgnFgvFcgnF^^ilF]^il>Figl-Fcy6#7'FagnFg^ilFcgnF^^ilF]^il-Fao6%%4(a )~The~solution~isGFbel%F~where~x1,~x3,~x5~are~free~variables|+G-Fao6%%4(b)~The~ solution~isGF[elF__il-Fao6%%4(c)~The~solution~isGFhelF__il-Fao6%%4(d)~The~solut ion~isGFiglF__ilFablC.Fggu>FdoFd]ilFgil-F?6#%]ois~the~augmented~matrix~of~a~non homogeneous~system~in~row~echelon|+G>F[el-Fcy6#7&Fagn,&F[oFFFcgnFgvFcgnFF>Fbel- Fcy6#7&FFhel-Fcy6#7&FFigl-Fcy6#7&Fagn,&F[oFFFcgnFF FcgnFF-Fao6%F^_ilFbel%E~where~x3~and~x4~are~free~variables|+G-Fao6%Fb_ilFigl%J~ where~x1,~x3,~and~x4~are~free~variables|+G-Fao6%Fe_ilFhel%>~where~x3~is~a~free~ variable|+G-Fao6%Fh_ilF[el%E~where~x1~and~x3~are~free~variables|+GF\^oC+FcgpFfg p>FdoF`bvFgil-F?6#%KThen~the~system~has~a~unique~solution~if~|+GF`]u-F?6#%0(b)~ a1-a2-a3=1|+GF^]hlFi]uFh^oC+FcgpFfgp>Fdo-Fio6#7%7&FFFbsFbjnFdtF]hp7&FFdo-Fio6#7%FjbilF]hp7&FFgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fed tC$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Fee qFcft>FdoFeftF7FfftF7, I,question1_1:FectFgctF7F7C)-FegrF[dt-F?6#%fn<~Enter~your~answer~below~OR~type~ exit;~to~quit~the~test~>|+G>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%FedtC%FhdtFC>FecqF \et@$F^etC%F`et>F]eqF^eq@%FedtC%FhdtFC>FecqF\etC'F[ftFC>F_ftFaft>FeeqFcft>FdoFe ftF7FfftF7, I,question1_2:FectFgctF7F7C)-F`]rF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$-Fao6#%GVERY~GOOD.~You~gave~the~correct~answerG>FecqF\et@$F^etC%F`et>F]eqF^eq @%FedtC$F[hil>FecqF\etC'F[ftFC>F_ftFaft>FeeqFcft>FdoFeftF7FfftF7, I,question1_3:FectFgctF7F7C)-F^[sF[dtF]fil>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fed tC$Ffet>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Fee qFcft>FdoFeftF7FfftF7, I,question1_4:FectFgctF7F7C)-Fg^sF[dtF\dt>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fedt C$Fhdt>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Feeq Fcft>FdoFeftF7FfftF7, I,question1_5:FectFgctF7F7C)-FfcrF[dtF]fil>FgeqF`dtFC>F]eqF^eq>FdoFF@$Faeq@%Fed tC$Ffet>FecqF\et@$F^etC%F`et>F]eqF^eq@%FedtC$Fhdt>FecqF\etC'F[ftFC>F_ftFaft>Fee qFcft>FdoFeftF7FfftF7, I(lpdquiz=F7FabqE\[l%%(quiz4_1G:F7F7F7F7C%-Fao6#%7Quiz~on~Inner~ProductsG-Fao6# %7----------------------G-Fj\_lFEF7F7%(quiz4_4G:F7F7F7F7C%-Fao6#%:Quiz~on~QR-De composition~G-Fao6#%:-------------------------G-Fj\_l6#F`rF7F7%(quiz4_2G:F7F7F7 F7C%-Fao6#%?Quiz~on~Orthogonal~ProjectionsG-Fao6#%?---------------------------- --G-Fj\_lFg]rF7F7%(quiz4_3G:F7F7F7F7C%-Fao6#%FgeqF`dtFC>F]eqF^eq>FdoFF@ $Faeq@%5Fedt/9&F]eqC%-F?6$%GVERY~GOOD.~(%c)~is~the~correct~answer|+GF]eqFC>Fecq F\et@$F^etC%F`et>F]eqF^eq@%F]ajlC%FaajlFC>FecqF\etC'F[ftFhbs>F_ftFaft>FeeqFcft> FdoFeftF7FfftF7