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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.

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  • $\begingroup$ This looks incomplete. Is anything missing? $\endgroup$ – PJK Apr 22 '17 at 3:43
  • $\begingroup$ Multiply the matrix $A$ with the vector (or column matrix)$(u,v,w)$. I think that's what has to be done. $\endgroup$ – астон вілла олоф мэллбэрг Apr 22 '17 at 3:51
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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*}

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  • $\begingroup$ Okay, so I multiply matrix A for L and get the rref of it? $\endgroup$ – Luzmaria Diaz Apr 22 '17 at 16:14
  • $\begingroup$ No, wait! I got it! Thank you! =) $\endgroup$ – Luzmaria Diaz Apr 22 '17 at 16:22

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