Math 220 Fall 2023 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 0 0 0 0 0 0 0 b 0 0 0 0 0 0 c 0 0 0 0 a 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 b v1 c v1 3 1 2 6 3 9 v3 12 v2 8 4 8 v2 12 v3 4 1 v2 0 v3 0 cid 20 3 2 cid 21 cid 20 3 2 2 6 3 9 1 3 0 0 0 1 1 0 7 compare your work and answers a 0 1 4 1 0 b 7 1 4 1 14 8 cid 21 4 Solve the following linear systems put your answers in parametric vector form then cid 20 3 2 c 7 3 9 1 4 1 cid 21 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 0 3 1 1 8 7 5 3 5 4 4 2 a 1 2 3 1 0 1 1 4 9 b 1 6 Suppose T R2 R2 is a linear transformation such that cid 18 cid 20 cid 21 cid 19 6 9 cid 21 cid 20 3 3 and T cid 18 cid 20 12 cid 21 cid 19 6 cid 20 0 18 cid 21 cid 18 cid 20 T cid 21 cid 19 6 7 Find T 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 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 2 radians counter clockwise 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 cid 21 cid 20 8 3 5 2 a b d 1 2 4 7 2 1 3 6 4 0 2 3 4 1 2 3 4 1 1 1 c 3 6 3 4 7 6 2 12 cid 20 1 2 cid 21 2 1 2 6 5 1 4 8 0 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 0 1 5 6 0 0 0 0 c 5 0 2 5 0 1 4 0 0 0 1 0 0 0 0 0 10 Find an LU factorization of A a geometric description of Nul A 6 5 1 12 3 4 8 3 4 5 9 2 1 4 8 3 7 1 2 7 4 2 2 9 5 3 6 9 5 2 3 5 a A b A 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 a b cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 3 0 1 2 2 1 1 0 1 1 1 3 2 1 0 2 3 1 1 4 1 1 1 1 1 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 12 Compute the determinant by any method 2 1 2 3 0 2 1 0 1 3 4 5 2 0 1 2 3 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 11 A matrix A and its echelon form are given below Find rank A and nullity A then give cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 d 2 3 4 0 1 0 1 0 1 0 0 0 3 0 0 0 2 2 5 1 4 2 0 4 1 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 13 Let A and B be 4 4 matrices with det A 3 and det B 1 Compute e det B 1AB c det 2A d det AT BA a det AB b det B5 14 Suppose that b c g a c cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 3d a 3e b 3f c c d e f g h i cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 a b cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 7 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 3a 3b cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 If so find the eigenvalue If so find the eigenvalue cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 3 2 6 7 Which of the following vectors are eigenvectors of A 3c 3d 3e 3f 3g 3h 3i 7 9 4 5 1 4 4 2 3 2 7 5 6 4 d h i Find each of the following determinants b d 3g e 3h f 3i a b c g a h b e cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 c d f 3g 3h 3i cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 a cid 12 cid 12 cid 12 cid 12 cid 12 cid 12 an eigenvector of 4 3 1 2 an eigenvector of 7 2 1 2 0 1 1 i 1 2 0 7 2 0 0 1 15 Is 16 Is 17 Let A a 1 0 1 c 1 1 0 d 1 1 1 b 0 1 1 1 0 1 Find the eigenvalues of A2 3 4 1 1 0 2 18 Find the characteristic equation of A 2 5 3 0 6 0 0 4 3 19 Let A 3 20 Find a basis for the eigenspace corresponding to each listed eigenvalue a A 4 4 2 cid 21 cid 20 10 9 3 1 3 4 2 cid 20 5 0 cid 21 3 1 1 3 9 2 4 3 3 6 2 1 6 2 1 4 3 b A c A cid 20 5 1 0 5 cid 21 cid 20 4 3 cid 21 b 1 cid 21 cid 20 2 …