Solved: Let B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} and B=

Chapter 6, Problem 17

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Let B = {(1, 1, 0), (1, 0, 1), (0, 1, 1)} and B’= {(1, 0, 0), (0, 1, 0), (0, 0, 1)} be bases for \(R^{3}\), and let

\(A=\left[\begin{array}{rrr} \frac{3}{2} & -1 & -\frac{1}{2} \\ -\frac{1}{2} & 2 & \frac{1}{2} \\ \frac{1}{2} & 1 & \frac{5}{2} \end{array}\right] \)

be the matrix for \(T: R^{3} \rightarrow R^{3}\) relative to B.

(a) Find the transition matrix P from B’ to B.

(b) Use the matrices P and A to find \([v]_{B}\) and \([T(v)]_{B}\), where \([v]_{B’} = [1 0 −1]^{T}\).

(c) Find \(P^{−1}\) and A’ (the matrix for T relative to B’).

(d) Find \([T(v)]_{B’}\) two ways.

Text Transcription:

R^3

A = [_1/2^-1/2^3/2 _1^2^-1 _5/2^1/2^-1/2]

T: R^3 rightarrow R^3

[v]_B

[T(v)]_B

[v]_B’ = [1 0 -1]^T

P^-1

[T(v)]_B’