Let A = {21,22,23) and B = {b,b2,63} be bases for a vector space V, and...
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 .
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.
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.)
Question 3 (10 marks) Suppose B-[bi, b2] and Cci, c2) are bases for a vector space V, even though we do not know the coordinates of all those vectors relative to the standard basis. However, we know that bi--c1 +3c2 and b2-2c1 -4c2 (a) Show that if C is a basis, then B is also a basis (b) Find N, given that x-5but 3b2. (c) Find lyle given that y Зе-5c2.
Question 3 (10 marks) Suppose B-[bi, b2] and Cci,...
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Find the new coordinate vector for the vector x after performing the specified change of basis. 8) Consider two bases B = 61, 62, 63 and C = C1, C2, C3) for a vector space V such that b1 = C1 + 2c3, b2 = C1 + 4c2 - C3, and b3 = 301 - C2. Suppose x = 51 +6b2 + b3. That is, 8) suppose [x]8 = 6 . Find [x]c A) B)
1. Let V be a vector space with bases B and C. Suppose that T:V V is a linear map with matrix representations Ms(T)A and Me(T) B. Prove the following (a) T is one-to-one iff A is one-to-one. (b) λ is an eigenvalue of T iff λ is an eigenvalue of B. Consequently, A and B have the same eigenvalues (c) There exists an invertible matrix V such that A-V-BV
1. Let V be a vector space with bases B...
QUESTION 2 Let B and B' be two bases of a vector space V, and let CHCMB-B. Choose the true statement. O The entry &of CHCMB' g is the ith coordinate in the basis B' of the ith vector of the basis B O The entry 0)of CHCMg' g is the ith coordinate in the basis B' of the ith vector of the basis B. O The entry &) of CHCMg' g is the ith coordinate in the basis B...
QUESTION 5 Let V denote an arbitrary finite-dimensional vector space with dimension n E N Let B = {bi, bn} and B' = { bị, b, } denote two bases for V and let PB-B, be the transition matrix from B to B' Prove that where 1 V → V is the identity transformation, i e 1(v) v for all v E V Note that I s a linear transformation 14]
QUESTION 5 Let V denote an arbitrary finite-dimensional vector...
3. Let V be a finite dimensional vector space with a positive definite scalar product. Let A: V-> V be a symmetric linear map. We say that A is positive definite if (Av, v) > 0 for all ve V and v 0. Prove: (a) if A is positive definite, then all eigenvalues are > 0. (b) If A is positive definite, then there exists a symmetric linear map B such that B2 = A and BA = AB. What...
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(10pts) Let B = {B1, B2, B3} be a basis for the vector space L2 x 2 of 2 x 2 lower triangular matrix, where B, = [:)] B -[i ), B =[ ] (a) Find the coordinate vector of Y = 2 07 with respect to B -13 (b) Find Z in L2x2 whose coordinates with respect to B is 2, 0,3