Discussion
If \(T : \mathbf{R}^{3} \to \mathbf{R}^{2}\) is given by :
\[ T\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x + z \\ y - z \end{pmatrix}, \]
then the matrix representation of T with respect to standard bases of \(\mathbf{R}^{3}\) and \(\mathbf{R}^{2}\) is :
Answer: B
The matrix is \(\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & -1 \end{pmatrix}\).
A map from \(\mathbf{R}^{3}\) to \(\mathbf{R}^{2}\) has a \(2 \times 3\) matrix whose columns are the images of the standard basis vectors:
\[ \begin{aligned} T(e_{1}) &= (1, 0) \\ T(e_{2}) &= (0, 1) \\ T(e_{3}) &= (1, -1) \end{aligned} \]
Reading the rows off \(x + z\) and \(y - z\) gives the same matrix. The \(3 \times 2\) options are transposes (they would map \(\mathbf{R}^{2}\) to \(\mathbf{R}^{3}\)), and the other \(2 \times 3\) option swaps the roles of \(x\) and \(y\).
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