I assume you mean:[tex]\displaystyle

\begin{align*}\\

T\begin{pmatrix}2\\3\end{pmatrix}&=\begin{pmatrix}-3\\3\end{pmatrix}\\

T\begin{pmatrix}3\\4\end{pmatrix}&=\begin{pmatrix} 5\\-4\end{pmatrix}

\end{align*}

[/tex]

Note: you should surround LaTeX code with [ t e x ]...[/ t e x ] (without the spaces) instead of $$...$$.

You should try to get the images of the unit vectors, as these correspond to the columns of the matrix.

For example, to get 0 in the second row, you would use the relation:.

[tex]\displaystyle

-4\begin{pmatrix}2\\3\end{pmatrix}+3\begin{pmatrix} 3\\4\end{pmatrix}=\begin{pmatrix}1\\0\end{pmatrix}

[/tex]

Note that, by luck, the first component is 1; in the general case, you may have to multiply both vectors by a constant to get a unit vector.

We have therefore:[tex]\displaystyle

\begin{align*}\\

T\begin{pmatrix}1\\0\end{pmatrix}&= -4\begin{pmatrix}-3\\3\end{pmatrix}+3\begin{pmatrix}5\\-4\end{pmatrix}\\

&= \begin{pmatrix}27\\-24\end{pmatrix}

\end{align*}

[/tex]

and this gives you the first column of the matrix. You can then use the same technique to get the second column.

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