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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,...
Question 2 (10 marks) Consider vectors b) (a) Show that B {bi, b2} and Ć = {ci. C2} are bases for R2 (b) Find the B-coordinates of x- (c) Find the change of coordinates matrix Pc-s from B to C and use it to find [x (d) Find the C-coordinates of y - (e) Find the change of coordinates matrix Psc from C to B and use it to find yg
Question 2 (10 marks) Consider vectors b) (a) Show...
Let A = {21,22,23) and B = {b,b2,63} be bases for a vector space V, and suppose a, = 5b, - b», a = -b + b + b3, az = b2 - 253 a Find the change-of-coordinates matrix from A to B. b. Find [x]g for x = 3a + 4a, +az. а PE BA b. [xlg (Simplify your answers.)
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.)
Let B = {bį, b2} and C = {C1,C2} be bases for R², where b, -6--0--0--01 1 a. Find P BEC [16 b. If [x]c = -3 de=[13] , find [x]
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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)
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 .
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(1 point) Consider the ordered bases B =( (8-4] [: • and c- (- -)( :} ) for the vector space V of lower triangular 2 x 2 matrices with zero trace. a. Find the transition matrix from C to B. TB = b. Find the coordinates of Min the ordered basis B if the coordinate vector of Min C is [Mc= [MB = C. Find M. M= (1 point) Consider the ordered bases B [ 1...
71121 4 1 15 Let S = -21,-1, 3 and C = | 1,2], o be bases for R'. Find the change-of- L5 JL-1] [4] IL-5] [ 8 7 ] coordinates matrix from B to C and the change-of-coordinates matrix from C to B. Find [x]. for x, = 4b, - 2b, +3b,. Find [x], for x, = 4c, - 2c, + 3e, Show these work by finding the coordinates of each vector in the standard basis using Band C
Find the matrix of the linear transformation T: V →W relative to B and C. Suppose B = {bı, b2, b3} is a basis for V and C = {C1, C2} is a basis for W. Let T be defined by T(b]) = 261 + C2 T(62) = -501 +502 T(b3) = 2C1-802 2. 3 0 2 -6 [3 0 -6 1 5-8 2 -5 2 5 -8 2 1 -5 5 2 -8