Specify the coefficient matrix for each of the following linear transformations:
- $L_1:\mathbb{R}^3\to \mathbb{R}^3$. $L_1(v)$ corresponds to the reflection of the vector $v=(x,y,z)$ at the $x,y$ plane.
- $L_2:\mathbb{R}^2\to\mathbb{R}^2$. $L_2(v)$ corresponds to the reflection of the vector $v$ at the line $G=\begin{pmatrix}1\\1\end{pmatrix}+\lambda \begin{pmatrix}1\\1\end{pmatrix}\; \lambda\in\mathbb{R}$
- $L_3:\mathbb{R}^2\to\mathbb{R}^2$. $L_3(v)$ corresponds to the rotation of the vector $v$ by the angle $\fracπ 4$ followed by the reflection on the straight line $G$ from b).
1.) We just need to invert $z$: $L(\begin{pmatrix}x\\y\\z\end{pmatrix})=\begin{pmatrix}x&0&0\\0&y&0\\0&0&-z\end{pmatrix}=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}\cdot \begin{pmatrix}x\\y\\z\end{pmatrix}$
2.) We need to find the images of the standardbasisvectors of $\mathbb{R}^2$, namely $(0,1),(1,0)$
$L(\begin{pmatrix}0\\1\end{pmatrix})=\begin{pmatrix}1\\0\end{pmatrix}$ and $L(\begin{pmatrix}1\\0\end{pmatrix})=\begin{pmatrix}0\\1\end{pmatrix}$
So it should be $L(\begin{pmatrix}v\end{pmatrix})=\begin{pmatrix}1&0\\0&1\end{pmatrix}$ or Am I wrong?
3.) Im not really sure about this one :(
in 2) the ordered base is (1,0) and (0,1) and the matrix would be $$ L_2=\begin{pmatrix} 0 & 1 \\1 & 0 \\\end{pmatrix}$$ you could use the other matrix but you would have to express your vectors in the form $(x,y)= y(0,1)+x(1,0)=\begin{pmatrix} y \\x \\\end{pmatrix}$ which is not very natural.
in 3) a rotation by $\pi/4$ send (1,0) to (0,1) and (0,1) to (-1,0) then a rotation have matrix $$ R=\begin{pmatrix} 0 & -1 \\1 & 0 \\\end{pmatrix}$$ finally $L_3=L_2\circ R$ , what corresponds to the multiplication of their respective matrices.