Linear Transformation using Standard Matrix

Question

Let $$A= \begin{bmatrix} -8 & 3 & 6\\ -3 & 1 & 2 \\ -3 & 1 & 3\\ \end{bmatrix} $$ be the standard matrix representing the linear transformation $$L: \Bbb R^3 \rightarrow \Bbb R^3 $$

Find $$L \left(\begin {bmatrix} u\\ v\\ w\\ \end{bmatrix}\right)$$ I have no idea where to even start. Can someone please explain.

I figured out the first part of the problem which was to find $$L: \left(\begin {bmatrix} 2\\ -3\\ 1\\ \end{bmatrix}\right) $$ which lead me to this answer: $$ \left(\begin {bmatrix} -19\\ -7\\ -6\\ \end{bmatrix}\right)$$ it is just the second part that I don't understand.

Answer 1

Suppose $T:\mathbb{R}^n\rightarrow \mathbb{R}^m$ is a linear transformation. Let $M$ be the standard matrix of $T$. This means that for any $u$ in $\mathbb{R}^n$, $Mu = T(u)$.

In your case, you want to find \begin{align*} L\begin{pmatrix}\begin{bmatrix}u\\v\\w\end{bmatrix}\end{pmatrix} = A\begin{bmatrix}u\\v\\w\end{bmatrix} = \begin{bmatrix} -8 & 3 & 6\\ -3 & 1 & 2 \\ -3 & 1 & 3\\ \end{bmatrix}\begin{bmatrix}u\\v\\w\end{bmatrix} \end{align*}