Sample Quiz on Linear Spaces Question # 1: The set of all 2x2 matrices of the form A=evalm(matrix([[a,0],[c,d]])); where a, b, and c are arbitrary real numbers, with respect to the usual addition and scalar multiplication of matrices: Select Answer Here (a) is not closed under addition (b) does not form a vector space (c) forms a vector space (d) is not closed under scalar multiplication Question # 2: Which statement is correct for every subset S of R3? If S Select Answer Here (a) includes the zero vector,then S a vector space (b) does not include the zero vector,then S is not a vector space (c) is not closed under addition,then S is a vector space (d) is not closed under scalar multiplication,then S is a vector space Question # 3: The set S consisting of all (x,y) such that x*y=1 is: Select Answer Here (a) an example of a vector space (b) an example of a set that is closed under addition (c) an example of a set that is closed under scalar multiplication (d) not an example of a vector space Question # 4: The minimun number of vectors needed to spans R3 is Select Answer Here (a) 3 (b) 4 (c) 2 (d) 1 Question # 5: The sets S1={(x1,x2)|x1+x2=0} and S2={(x1,x2)|x1-x2=0}. Then Select Answer Here (a) The intersection of S1 and S2 is a subspace spanned by one vector 1 (b) The intersection of S1 and S2 is a subspace of dimension 0 (c) The union of S1 and S2 is a subspace spanned by one vector (d) The union of S1 and S2 is a subspace spanned by two vectors Question # 6: The set S={ (x1,x2,x3)| x1+x2-x3=1} Select Answer Here (a) S is a subspace spanned by two vectors (b) S is not a subspace because it is closed under addition but closed under scalar multiplication (c) S is not a subspace because it is not closed under addition but closed under scalar multiplication (d) S is not a subspace because the zero element is not in S Question # 7: The set { (x1,x2,x3)| x1+x2=x3} Select Answer Here (a) is an example of a subspace of dimension 1 (b) is an example of a subspace of dimension 2 (c) is an example of a subspace of R2 (d) is an example of a subspace containing the point (1,1,3) Question # 8: A basis for the row space of the matrix A A:=matrix([[1,0,1,1],[2,0,2,2],[-1,0,-1,-1]]) Select Answer Here (a) {[1,0,1,1], [-1,0,-1,-1]} (b) {[1,0,1,1]} (c) {[1,0,1,1],[2,0,2,2]} (d) {[1,0,1,1],[2,0,2,2],[-1,0,-1,-1]}