Sample Quiz on Linear Transformations Question # 1: If B= {v1,v2,v3} is a basis for the vector space R3 and T is a one-to-one and onto linear transformation from R3 to R3, then Select Answer Here (a) T(B) is a linearly dependent set (b) T(B) is not a basis for R3 (c) T(B) is a basis for R3 (d) T(B) does not span R3 Question # 2: If T: V --> W is a linear transformation and ker(T)={0}, then Select Answer Here (a) T is a one-to-one mapping (b) T is a one-to-one and onto mapping (c) T is not a one-to-one mapping (d) T is not a one-to-one mapping but onto Question # 3: If T: R3 --> R2 is defined by T(x1,x2,x3) = (x1-x2,x3), then Select Answer Here (a) a basis for ker(T) is {(0,0,0)} (b) a basis for ker(T) is {(1,1,0)} (c) a basis for ker(T) is {(1,1,0),(0,0,1)}} (d) a basis for ker(T) is {(0,0,1)} Question # 4: If T: R3 --> R2 is defined by T(x1,x2,x3) = (x1-x2,x3), then Select Answer Here (a) the range(T) is a proper subspace of R2 (b) the range(T) is not a subspace of R3 (c) the range(T) is R2 (d) the range(T) is R3 Question # 5: A 5x4 matrix A represents a linear transformation Select Answer Here (a) from R5 to R4 (b) which is one-to-one (c) which is onto (d) from R4 to R5 Question # 6: If T: R3 --> R2 is defined on the standard basis as follows: T(1,0,0) = (1,1) ; T(0,1,0) = (2,1) ; T(0,0,1) = (-1,1) then the transformation T is given by: Select Answer Here (a) T(x1,x2,x3) = (x1+2x2-x3, x1+x2+x3) (b) T(x1,x2,x3) = (x1+2x2-x3, -x1+x2+x3) (c) T(x1,x2,x3) = (x1+2x2-x3, x1+x2-x3) (d) T(x1,x2,x3) = (x1+2x2+x3, x1+x2+x3) Question # 7: If a basis for ker(T) is {(1,1,1)} and a basis for range(T) is {(1,0,1), (-1,1,0)} for T: R3 --> R3 , then the solution to the nonhomogenous system of equations Ax=b with A being the matrix representation of T: Select Answer Here (a) is k1*(1,1,1) + span(range(T)) and k1 is any real number (b) is k1*(1,1,1) and k1 is an arbitrary real number (c) consists of span(range(T) (d) is k1*(1,1,1) + (0,1,1) and k1 is any real number Question # 8: If T: R2 --> R2 is defined by the matrix multiplication where the matrix A is A=matrix([[1,-1],[1,0]]). Then the inverse of transformation T is the transformation T1: Select Answer Here (a) T1:= (x1-x2,x2) (b) T1:= (x1,x1-x2) (c) T1:= (-x1+x2,x2) (d) T1:= (x2,-x1+x2)