Determine $f(v)$ and $f(w)$ in the coordinate represention in $\mathbb{R}^3$ and the mapping matrix $M^{E_4}_{E_3}(f)$

Question

Given is the linear mapping $f:\mathbb{R}^4\to \mathbb{R}^3$ with $$M_\mathcal{B}^\mathcal{A}(f)=\begin{pmatrix}4 & 0 & -4 & 2 \\ 0 & 2 & 4 & 2 \\ -2 & 0 & 0 & -2 \end{pmatrix}$$ with respect to the bases $$\mathcal{A}=(\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix},\begin{pmatrix} 0\\ 1 \\ 1 \\ 0 \end{pmatrix},\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix},\begin{pmatrix} 0 \\ -1 \\ 1 \\ 0 \end{pmatrix})\quad \mathcal{B}=(\begin{pmatrix} 1\\1\\1 \end{pmatrix},\begin{pmatrix} 1\\0\\1 \end{pmatrix},\begin{pmatrix} 1\\2\\2 \end{pmatrix}$$

a) The vectors $v=\begin{pmatrix} 2\\2\\-2\\2 \end{pmatrix}$ and $w=\begin{pmatrix} -1\\1\\0\\1 \end{pmatrix}$ are given. Using the matrix $M^\mathcal{A}_\mathcal{B}(f)$, determine the vectors $f(v)$ and $f(w)$ in the coordinate representation with respect to the standard basis of $\mathbb{R}^3$.

b) Let $E_4$ and $E_3$ be the standard bases of $\mathbb{R}^4$ and $\mathbb{R}^3$, respectively. Determine the mapping matrix $M^{E_4}_{E_3}(f)$ and check your previously calculated results with the matrix.

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Can you give me hints on how to solve the two tasks?

Answer 1

For (a), just compute $[v]_{\mathcal{A}}$ and $[w]_{\mathcal{A}}$ (the coordinate vectors of $v$ and $w$ respect to the basis $\mathcal{A}$). Then $$[f(v)]_{\mathcal{B}}=\textsf{M}_{\mathcal{B}}^{\mathcal{A}}(f)[v]_{\mathcal{A}}$$ and similarly $$[f(w)]_{\mathcal{B}}=\textsf{M}_{\mathcal{B}}^{\mathcal{A}}(f)[w]_{\mathcal{A}}$$ For example, for $v$ we have $$v=\begin{pmatrix}2\\2\\-2\\2\end{pmatrix}=2\begin{pmatrix}1\\0\\0\\1\end{pmatrix}+0\begin{pmatrix}0\\1\\1\\0\end{pmatrix}+0\begin{pmatrix}1\\0\\0\\-1\end{pmatrix}-2\begin{pmatrix}0\\-1\\1\\0\end{pmatrix}$$ hence $$[v]_\mathcal{A}=\begin{pmatrix}2\\0\\0\\-2\end{pmatrix}$$ and then $$[f(u)]_\mathcal{B}=\begin{pmatrix}4&0&-4&2\\0&2&4&2\\-2&0&0&-2\end{pmatrix}\begin{pmatrix}2\\0\\0\\-2\end{pmatrix}=\begin{pmatrix}4\\-4\\0\end{pmatrix}$$ Therefore $$f(u)=4\begin{pmatrix}1\\1\\1\end{pmatrix}-4\begin{pmatrix}1\\0\\1\end{pmatrix}+0\begin{pmatrix}1\\2\\2\end{pmatrix}=\begin{pmatrix}0\\4\\0\end{pmatrix}$$ Can you continue?

Answer 2

For a more systematic approach:

$$ M^{E_4}_{E_3}(f)=M^{E_4}_{E_3}(id\circ f\circ id)=M^{B}_{E_3}(id)\cdot M^{A}_{B}(f)\cdot M^{E_4}_{A}(id)\\=M^{B}_{E_3}(id)\cdot M^{A}_{B}(f)\cdot (M^{A}_{E_4}(id))^{-1} $$

Now $M^{B}_{E_3}(id)$, $M^{A}_{E_4}(id)$ and $M^{A}_{B}(f)$ are already given so you can happily compute $M^{E_4}_{E_3}(f)$ and $[f(u)]_{E_3}=M^{E_4}_{E_3}(f)\cdot [u]_{E_4}$.