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Ok, so this is a practice question in my book:

$V$ is $M_{22}$

$S=$ \begin{bmatrix} 1&-2\\ 0&0\\ \end{bmatrix} \begin{bmatrix} -1&3\\ 0&1\\ \end{bmatrix} \begin{bmatrix} 1&0\\ 0&0\\ \end{bmatrix} \begin{bmatrix} 0&-1\\ 1&0\\ \end{bmatrix}

and $[v]_{s}=$

\begin{bmatrix} 0\\ 1\\ 0\\ 2\\ \end{bmatrix}

The answer is \begin{bmatrix} -1&1\\ 2&1\\ \end{bmatrix}

But I can't figure out how they get to that. I know it is supposed to be

$[v]_{s}=$ \begin{bmatrix} a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\\ \end{bmatrix}

But I cant figure how they get to it. The 2x2 matrix syntax is confusing me.

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Ok, so I had a friend explain it to me!

Here is the answer:

Since $[v]_{s}=$

\begin{bmatrix} a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\\ \end{bmatrix}

you just multiply $a_{1}* v_{1} + a_{2}*v_{2} + a_{3}* v_{3}+a_{4}* v_{4}$ to get your answer.

Very simple really.

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