Exercise 5 Let 2 1 -2 -1 (a) Find A4 (b) Find eA (c) Find a matrix B such that B2 = A. Exercise 5 Let 2 1 -2 -1 (a...
Let B = {b1,b2} and C= {(1,62} be bases for R2. Find the change-of-coordinates matrix from B to C and the change-of-coordinates matrix from C to B. - 1 b = b2 = C1 = C = 4 -3 Find the change-of-coordinates matrix from B to C. P = CB (Simplify your answers.) Find the change-of-coordinates matrix from C to B. P B-C [8: (Simplify your answers.)
Exercise 24. Let 2 1 A =-1 3 1 0 -2 2 3 SDS-1. (i) Find a nonsingular matrix S and a diagonal matrix D such that A (ii) Find a matrix B that satisfies B2 = A Exercise 24. Let 2 1 A =-1 3 1 0 -2 2 3 SDS-1. (i) Find a nonsingular matrix S and a diagonal matrix D such that A (ii) Find a matrix B that satisfies B2 = A
Exercise 30. Let A be a 5 x 5 matrix. Find the Jordan canonical form J under each of the following assumptions (i) A has only eigenvalue namely 4 and dim N(A- 41) = 4. one (ii) dim N(A 21) = 5. (ii dim N(A -I) = 3 and dim N (A 31) 2. (iv) det(A I) = (1 - )2(2 - A)2 (3 - ) and dim N(A - I) dim N(A - 21) 1 (v) A5 0 and...
Assume that the transition matrix from basis B = {b1, b2, b3} to basis C = {c1, c2, c3} is PC,B = 1/2*[ 0 -1 1 ; -1 1 1 ; 1 0 0 ]. (a) If u = b1 + b2 + 2b3, find [u]C. (b) Calculate PB,C. (c) Suppose that c1 = (1, 2, 3), c2 = (1, 2, 0), c3 = (1, 0, 0) and let S be the standard basis for R 3 . (i) Find...
Question 6. (15 pts) Let B = {bı, b2} and C = {ci, c2} be bases for a vector space, and suppose bı = - + 4c2 and b2 = 501 - 3c2. (1). Find the change-of-coordinates matrix from B to C. (2). Find [x]c for x = 5bı + 3b2.
Question 2. Let 1 -15 B = 1 1 2 V2 a) Compute B2, B3, B4, B7, and B8. b) Use part a) to determine B2020. Show your work. c) The matrix B is invertible. Use part a) to find B-1. Justify your answer. (Note: no marks will be given if either the formula for the inverse of a 2 x 2 matrix or row reduction is used to compute B-1)
Exercise 1. (a) Find the inverse of the matrix 0 0 1/2 A= 01/ 31 1/5 1 0 (b) Let N be a nxn matrix with N2 = 0. Show (I. - N)-1 = IA+N. (Hint: Use the definition of the inverse.)
1. Let B and B' be the bases an 000) (a) Find the transition matrix Pta-B from the standard basis {e, e2, ea) to B (b) Find the transition matrix PB-B from B to B' (c) Use (a) to find wlB where in terms of the standard basis (d) Use (b) and your previous answer to find [w]B' (e) Make a conjecture about how we should interpret PB-B Pstan-B
2. (a) Find a 2 x 2 matrix A such that AP + 12 = 0. (b) Show that there is no 5 x 5 matrix B such that B2 + 15 = 0. (c) Let C be any n xn matrix such that C2 + In 0. Let l be any eigenvalue of C. Show that 12 Conclude that C has no real eigenvalues. [1] [3] =-1. [3]
Let B = {b1,b2, b3} be a basis for a vector space V. Let T be a linear transformation from V to V whose matrix relative to B is [ 1 -1 0 1 [T]B = 2 -2 -1 . 10 -1 -3 1 Find T(-3b1 – b2 - b3) in terms of bı, b2, b3 .