## Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known

## Problem 159

Let $T: \R^2 \to \R^2$ be a linear transformation.

Let

\[

\mathbf{u}=\begin{bmatrix}

1 \\

2

\end{bmatrix}, \mathbf{v}=\begin{bmatrix}

3 \\

5

\end{bmatrix}\]
be 2-dimensional vectors.

Suppose that

\begin{align*}

T(\mathbf{u})&=T\left( \begin{bmatrix}

1 \\

2

\end{bmatrix} \right)=\begin{bmatrix}

-3 \\

5

\end{bmatrix},\\

T(\mathbf{v})&=T\left(\begin{bmatrix}

3 \\

5

\end{bmatrix}\right)=\begin{bmatrix}

7 \\

1

\end{bmatrix}.

\end{align*}

Let $\mathbf{w}=\begin{bmatrix}

x \\

y

\end{bmatrix}\in \R^2$.

Find the formula for $T(\mathbf{w})$ in terms of $x$ and $y$.