Answer to Question #211217 in Linear Algebra for Sani

Question

Let T : R3 → R3 be defined by T (x1

, x2

, x3

) = (x1

, x2

,−x1 − x2

). Find a

matrix which represents T

Answer 1

Let $T:R^3\to R^3$ be defined by $T(x_1, x_2, x_3)=(x_1, x_2, -x_1-x_2)$

$$T\begin{pmatrix} \begin{bmatrix} x_1\\ x_2 \\ x_3 \end{bmatrix} \end{pmatrix}=\begin{bmatrix} x_1 \\ x_2\\ -x_1-x_2 \end{bmatrix}$$

$$T\begin{pmatrix} \begin{bmatrix} 1\\ 0 \\ 0 \end{bmatrix} \end{pmatrix}=\begin{bmatrix} 1 \\ 0\\ -1 \end{bmatrix}$$

$$T\begin{pmatrix} \begin{bmatrix} 0\\ 1 \\ 0 \end{bmatrix} \end{pmatrix}=\begin{bmatrix} 0 \\ 1\\ -1 \end{bmatrix}$$

$$T\begin{pmatrix} \begin{bmatrix} 0\\ 0 \\ 1 \end{bmatrix} \end{pmatrix}=\begin{bmatrix} 0\\ 0\\ 0 \end{bmatrix}$$

Using these as our columns, the standard matrix for $T$ is:

$$A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -1 & -1 & 0 \\ \end{bmatrix}$$