## Find a Value of a Linear Transformation From $\R^2$ to $\R^3$

## Problem 142

Let $T:\R^2 \to \R^3$ be a linear transformation such that $T(\mathbf{e}_1)=\mathbf{u}_1$ and $T(\mathbf{e}_2)=\mathbf{u}_2$, where $\mathbf{e}_1=\begin{bmatrix}

1 \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1

\end{bmatrix}$ are unit vectors of $\R^2$ and

\[\mathbf{u}_1= \begin{bmatrix}

-1 \\

0 \\

1

\end{bmatrix}, \quad \mathbf{u}_2=\begin{bmatrix}

2 \\

1 \\

0

\end{bmatrix}.\]
Then find $T\left(\begin{bmatrix}

3 \\

-2

\end{bmatrix}\right)$.