## Restriction of a Linear Transformation on the x-z Plane is a Linear Transformation

## Problem 428

Let $T:\R^3 \to \R^3$ be a linear transformation and suppose that its matrix representation with respect to the standard basis is given by the matrix

\[A=\begin{bmatrix}

1 & 0 & 2 \\

0 &3 &0 \\

4 & 0 & 5

\end{bmatrix}.\]

**(a)** Prove that the linear transformation $T$ sends points on the $x$-$z$ plane to points on the $x$-$z$ plane.

**(b)** Prove that the restriction of $T$ on the $x$-$z$ plane is a linear transformation.

**(c)** Find the matrix representation of the linear transformation obtained in part (b) with respect to the standard basis

\[\left\{\, \begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}, \begin{bmatrix}

0 \\

0 \\

1

\end{bmatrix} \,\right\}\]
of the $x$-$z$ plane.