Let \(T\) be the linear transformation from \(\RR^3\) into \(\RR^2\) defined by \[ T(x_1, x_2, x_3) = (x_1 + x_2,\; 2x_3 - x_1). \]
If \(\mathfrak{B}\) is the standard ordered basis for \(\RR^3\) and \(\mathfrak{B}'\) is the standard ordered basis for \(\RR^2\), what is the matrix of \(T\) relative to the pair \(\mathfrak{B}\), \(\mathfrak{B}'\)?
If \(\mathfrak{B} = \{\alpha_1, \alpha_2, \alpha_3\}\) and \(\mathfrak{B}' = \{\beta_1, \beta_2\}\), where \[ \alpha_1 = (1, 0, -1), \quad \alpha_2 = (1, 1, 1), \quad \alpha_3 = (1, 0, 0), \quad \beta_1 = (0, 1), \quad \beta_2 = (1, 0) \] what is the matrix of \(T\) relative to the pair \(\mathfrak{B}\), \(\mathfrak{B}'\)?
Solution
In the standard bases, \[ [T]=\begin{bmatrix}1&1&0\\-1&0&2\end{bmatrix}, \] since these are the coordinate columns of \(T(e_1),T(e_2),T(e_3)\).
For part (b), let \[ P=\begin{bmatrix}1&1&1\\0&1&0\\-1&1&0\end{bmatrix} \] be the domain change-of-basis matrix, and \[ Q=\begin{bmatrix}0&1\\1&0\end{bmatrix} \] be the codomain change-of-basis matrix. Then \[ [T]_{\mathfrak B,\mathfrak B'}=Q^{-1}[T]P =\begin{bmatrix}-3&1&-1\\1&2&1\end{bmatrix}. \]