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Solution to Linear Algebra Done Wrong Exercise 1.3.4


Find $3\times 3$ matrices representing the transformations of $\mathbb R^3$ which:
a) project every vector onto $x\text{-}y$ plane;
b) reflect every vector through $x\text{-}y$ plane;
c) rotate the $x\text{-}y$ plane through $30^\circ$, leaving $z$-axis alone.


Solution:

a) It is clear this transformation $T$ maps $(x,y,z)^T$ to $(x,y,0)^T$. We have $$T\begin{pmatrix}x\\ y\\ z\end{pmatrix}=x\begin{pmatrix}1\\ 0 \\ 0\end{pmatrix}+y\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}+z\begin{pmatrix}0\\ 0\\ 0\end{pmatrix}.$$Therefore, the matrix is $\begin{pmatrix} 1 & 0 & 0\\ 0 & 1& 0\\ 0& 0 &0\end{pmatrix}$.

b) It is clear this transformation $T$ maps $(x,y,z)^T$ to $(x,y,-z)^T$. We have $$T\begin{pmatrix}x\\ y\\ z\end{pmatrix}=x\begin{pmatrix}1\\ 0\\ 0\end{pmatrix}+y\begin{pmatrix}0\\ 1\\ 0\end{pmatrix}+z\begin{pmatrix}0\\ 0\\ -1\end{pmatrix}.$$Therefore, the matrix is $\begin{pmatrix} 1 & 0 & 0\\ 0 & 1& 0\\ 0 & 0 & -1\end{pmatrix}$.

c) The transformation changes $x$ and $y$ but fix $z$. Similar to the above, the matrix is either $$\begin{pmatrix} \cos 30^\circ & \sin 30^\circ & 0\\ -\sin 30^\circ &\cos 30^\circ & 0\\ 0 &0 &1\end{pmatrix}$$ or $$\begin{pmatrix} \cos 30^\circ & -\sin 30^\circ & 0\\ \sin 30^\circ &\cos 30^\circ & 0\\ 0 &0 &1\end{pmatrix}$$ depending on the direction of rotation, see Page 21.

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