Math 220 Spring 2022 Final Exam Review 1 The matrices given below are in echelon form with representing a nonzero entry Each augmented matrix represents a linear system Determine if the system is consistent If it is consistent determine if the solution is unique a b c 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 Consider the system x1 hx2 2 4x1 8x2 k Choose h and k so that the system has a no solution b a unique solution c many solutions 3 Give a geometric description of Span v1 v2 v3 a v1 v2 v3 8 2 6 12 3 9 b v1 v3 v2 8 2 6 1 0 v2 0 12 3 9 0 1 v3 0 0 0 1 c v1 4 3 1 4 1 3 1 4 Solve the following vector equations put your answers in parametric vector form then compare your work and answers 5 Determine if each set is linearly independent If the set is linearly dependent give an example of how the third vector can be written as a linear combination of the other vectors Then give an example of how the second vector can be written as a linear combination of the other vectors a b c cid 20 3 2 7 0 1 4 1 0 cid 21 cid 20 3 2 7 1 4 1 14 8 cid 20 3 2 7 3 9 1 4 1 cid 21 cid 21 0 3 1 1 1 2 3 8 7 5 3 1 0 1 1 4 9 5 4 4 2 a b 6 Suppose T R2 R2 is a linear transformation such that cid 21 cid 19 cid 18 cid 20 5 2 T cid 21 cid 20 2 1 and T cid 21 cid 19 cid 18 cid 20 1 3 cid 21 cid 20 1 3 Find T cid 18 cid 20 10 9 cid 21 cid 19 7 Assume that T is a linear transformation Find the standard matrix of T a T R2 R2 rotates points about the origin through 3 2 radians counter clockwise b T R2 R2 reflects points across the x axis then reflects across the line y x c T R3 R3 projects onto the xz plane d T R3 R2 defined by T x1 x2 x3 x1 5x2 4x3 x2 6x3 8 Is the transformation in 7 d one to one Is it onto 9 Find the inverse of each matrix or determine the matrix is not invertible a b cid 21 cid 21 cid 20 8 3 5 2 cid 20 1 2 3 6 2 10 Determine which sets are a basis for R2 cid 21 cid 27 0 2 1 4 3 1 4 2 3 1 2 4 7 2 1 3 6 4 cid 20 10 3 cid 20 2 3 cid 20 10 3 cid 21 cid 27 cid 20 0 1 cid 21 cid 21 cid 26 cid 20 5 2 cid 21 cid 26 cid 20 4 6 cid 26 cid 20 5 2 cid 21 cid 26 cid 20 1 0 cid 26 cid 20 1 cid 21 cid 27 0 c d a b c d e cid 21 cid 27 cid 21 cid 27 cid 21 cid 20 1 1 whenever possible a H x 0 cid 27 cid 21 cid 26 cid 20 x y cid 26 cid 20 x 2 cid 21 cid 26 cid 20 x x cid 26 cid 20 x y cid 21 cid 21 cid 27 x R cid 27 x R cid 27 b H c H d H y x 3 11 A matrix A and its echelon form are given below Find a basis for Col A and N ul A a A b A 4 5 9 2 6 5 1 12 3 4 8 3 1 2 6 5 0 1 5 6 0 0 0 0 1 1 2 7 2 2 9 3 4 8 3 7 3 4 5 5 6 9 5 2 1 4 8 0 5 0 2 5 0 1 4 0 0 0 1 0 0 0 0 0 12 Which of the following subsets of R2 are subspaces Give a geometric interpretation 1 3 2 4 3 9 6 12 2 1 4 2 4 5 3 7 14 The vector x is in a subspace H with a basis B b1 b2 Find the B coordinate 13 Find a basis for the subspace spanned by the vectors What is the dimension of the subspace vector of x a b1 b2 x cid 21 cid 20 1 4 1 5 3 cid 21 cid 20 2 7 3 7 5 cid 21 cid 20 3 7 4 10 7 b b1 b2 x 15 Compute the determinant by cofactor expansion a b c d e a b cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 4 3 0 2 3 2 0 5 1 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 3 2 2 1 3 2 3 1 1 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 4 0 0 5 1 7 2 5 0 3 0 0 7 8 3 1 5 6 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 3 0 2 0 0 0 0 4 3 3 5 1 3 0 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 4 0 7 0 0 2 7 3 6 5 5 0 9 1 0 0 3 5 0 0 4 8 2 3 2 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 1 5 4 5 7 1 4 2 8 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 1 2 3 0 4 2 5 7 3 1 5 2 1 1 2 3 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 4 16 Find the determinant by row reduction to echelon form 17 Let A and B be 4 4 matrices with det A 3 and det B 1 Compute a det AB b det B5 c det 2A d det AT BA e det B 1AB 18 Suppose that cid 12 cid 12 cid 12 cid 12 cid 12 cid …