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.
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