- #1

fatpotato

- Homework Statement
- Change of basis in matrix space : Find the coordinate matrix for 2 by 2 matrix ##A## relative to set ##S = \{A_1, A_2, A_3, A_4\}##

- Relevant Equations
- Matrix ## A = \begin{bmatrix} 2 & 0 \\-1 & 3 \end{bmatrix}##

Set ##S = \{A_1 = \begin{bmatrix} -1 & 1 \\0 & 0 \end{bmatrix}, A_2 = \begin{bmatrix} 1 & 1 \\0 & 0 \end{bmatrix}, A_3 = \begin{bmatrix} 0 & 0 \\1 & 0 \end{bmatrix}, A_4 = \begin{bmatrix} 0 & 0 \\0 & 1 \end{bmatrix} \}##

Hello,

I am studying change of basis in linear algebra and I have trouble figuring what my result should look like.

From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##, giving me an ordered list ##a = \{-1,1,-1,3\}## that I can match with each basis vector of ##S## to obtain ##A##.

However, in the context of change of basis, I can't comprehend what we expect the result to be. I have dealt with straightforward problems where the task was to find a transition matrix in ##\mathbb{R}^2## or ##\mathbb{R}^3##, so in the end we would get a matrix and its inverse to use as a mean for translating one vector from one representation to another.

In this case, I am not sure the list ##a## should be written in matrix form...How should I interpret this result?

Edit : error in list ##a## and solution

I am studying change of basis in linear algebra and I have trouble figuring what my result should look like.

From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##, giving me an ordered list ##a = \{-1,1,-1,3\}## that I can match with each basis vector of ##S## to obtain ##A##.

However, in the context of change of basis, I can't comprehend what we expect the result to be. I have dealt with straightforward problems where the task was to find a transition matrix in ##\mathbb{R}^2## or ##\mathbb{R}^3##, so in the end we would get a matrix and its inverse to use as a mean for translating one vector from one representation to another.

In this case, I am not sure the list ##a## should be written in matrix form...How should I interpret this result?

Edit : error in list ##a## and solution

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