Vector Transformation

Question

I have $T$ defined by $T \textbf{x}=(\textbf{x} \cdot(3,2,1))(3,2,1)+(\textbf{x} \cdot(-1,0,-2))(-1,0,-2) $

and the following questions:

I multiplied out the brackets to find the image $i=\begin{bmatrix}10 \\ 0 \\ 0 \end{bmatrix}$, $j=\begin{bmatrix}0 \\ 4\\ 0 \end{bmatrix}$, $k=\begin{bmatrix}0 \\ 0 \\ 5 \end{bmatrix}$ and standard matrix representation being $\begin{bmatrix}10 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 5\end{bmatrix}$ however I'm not sure if this right as I haven't encountered problems set out in that way.

Answer 1

I suppose that $\{\mathbf{i},\mathbf{j},\mathbf{k}\}$ is the standard basis of $\mathbb{R}^3$.

Note that

• $T(\mathbf{i})=3(3,2,1)-(-1,0,-2)=(10,6,5)$;
• $T(\mathbf{j})=2(3,2,1)=(6,4,2)$;
• $T(\mathbf{k})=(3,2,1)-2(-1,0,-2)=(5,2,5)$.

So,$$A_T=\begin{bmatrix}10&6&5\\6&4&2\\5&2&5\end{bmatrix}.$$