Change of basis to express a matrix relative to a set of basis matrices

[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

 

[fresh_42]

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

If $\{b_1,\ldots,b_4\}$ is a basis of a $4-$dimensional real vector space $V$ and $v\in V$ a vector, then there are $\lambda_1,\ldots,\lambda_4\in \mathbb{R}$ such that $v=\lambda_1b_1+\ldots+\lambda_4b_4$. Your coordinate vector $a$ would be $(\lambda_1,\ldots,\lambda_4)$ in this case.

That's it. The only difference is, that you have all vectors given in a different notation, namely matrices. How you interpret $a$ depends on what you want to do with it. $a$ is simply the coordinates according to the basis $b_1=A_1,\ldots,b_4=A_4.$ There is nothing more to it.

 

[fatpotato]

Thank you very much for your response.

So in this case, there would not be any convenient way of expressing a change of basis like the matrix multiplication I gave with a computational goal in mind? For example, matrix computation is straight forwardly implemented in numerous programming languages.

 

[fresh_42]

The set of all matrices over a field is a vector space, and that is all what it is here. If the matrices are square, then it is even an algebra or ring where you can multiply, but this is another subject. If the square matrices are regular, then it is no longer a vector space but a multiplicative group instead, but this is also not requested here.

Hence matrix multiplication may come into play ...

How you interpret $a$ depends on what you want to do with it.

... but not within the context you provided. The only goal was to learn that matrices form a vector space.

 

[Stephen Tashi]

Homework Statement:: Change of basis in matrix space

I am studying change of basis in linear algebra

It's worth being mentally prepared to work a different type of "change of basis" involving matrices, so you won't confuse it with the type of problem in your original post. An NxN matrix can be considered to define a linear mapping of an N-dimensional vector space into itself relative to a particular basis for that N-dimensional vector space. Eventually you will encounter problems that ask you re-write the matrix A that is expressed in terms of one basis for the N dimensional vector space as a different matrix B that is expressed in term of a different basis for the N-dimensional vector space.