Given V = R, and two bases: B and C and the coordinator [v]B a) Find the change of coordinates matrix PB rightarrow C. b) Find the coordinator [v]C.

Given V = R, and two bases: B and C and the coordinator [v]B a) Find the change of coordinates matrix PB rightarrow C. b) Find the coordinator [v]C.

Question
Alternate coordinate systems
asked 2021-02-19
Given V = R, and two bases: B and C and the coordinator [v]B a) Find the change of coordinates matrix \(\displaystyle{P}{B}\rightarrow{C}.\) b) Find the coordinator [v]C.

Answers (1)

2021-02-20

a .\(\begin{bmatrix} 9 \\ 2 \end{bmatrix} = \prec_1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} + \prec_2 \begin{bmatrix} -3 \\ 1 \end{bmatrix} \Rightarrow 2 \prec_1 + (-3) \prec_2 = 9 \prec_1 + \prec_2 = 2 \Rightarrow 2 \prec_1 + 2 \prec_2 = 4 2 \prec_1 + -3 \prec_2 = 9 \prec_2 = -1 \Rightarrow \prec_1 = 3 \begin{bmatrix} 4 \\ -3 \end{bmatrix} = B1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} + B2 \begin{bmatrix} -3 \\ 1 \end{bmatrix} \Rightarrow 2 B1 - 3 B2 = 4 B1 + B2 = -3 \Rightarrow 2 B1 + 2 B2 = -6 2 B1 - 3 B2 = 4 \Rightarrow B2 = -2, B1 = -1 P_{B \leftarrow C} = \begin{bmatrix} \prec_1 & B1 \\ \prec_2 & B2 \end{bmatrix} = \begin{bmatrix} 3 & -1 \\ -1 & -2 \end{bmatrix}\)

b. \(​ Now \ v = \begin{bmatrix} 34 \\ 27 \end{bmatrix} = \curlyvee _1 \begin{bmatrix} 2 \\ 1 \end{bmatrix} \curlyvee_2 \begin{bmatrix} -3 \\ 1 \end{bmatrix} \Rightarrow 2 \curlyvee_1 - 3 \curlyvee_2 = 34 \curlyvee_1 + \curlyvee_2 = 27 \Rightarrow 2 \curlyvee_1 + 2 \curlyvee_2 = 54 \Rightarrow - 5 \curlyvee_2 = -20 \Rightarrow \curlyvee_2 = 4 \Rightarrow \curlyvee_1 = 23 \Rightarrow [v]C = \begin{bmatrix} \curlyvee_1 \\ \curlyvee_2 \end{bmatrix} = \begin{bmatrix} 23 \\ 4 \end{bmatrix}\)

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