## Give the Formula for a Linear Transformation from $\R^3$ to $\R^2$

## Problem 156

Let $T: \R^3 \to \R^2$ be a linear transformation such that

\[T(\mathbf{e}_1)=\begin{bmatrix}

1 \\

4

\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}

2 \\

5

\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}

3 \\

6

\end{bmatrix},\]
where

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

1 \\

0 \\

0

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

0 \\

1 \\

0

\end{bmatrix}, \mathbf{e}_3=\begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix}\]
are the standard unit basis vectors of $\R^3$.

For any vector $\mathbf{x}=\begin{bmatrix}

x_1 \\

x_2 \\

x_3

\end{bmatrix}\in \R^3$, find a formula for $T(\mathbf{x})$.