Exercise 1.5.4

Question

Find the matrix of the orthogonal projection in $\mathbb{R}^2$ onto the line $x_1 = -2x_2$.

Jing Huang · Accepted Answer

extbf{Solution}
\begin{equation*}
  \begin{split}
    T &= 
    R(\alpha)PR(-\alpha) \
    &=
    R(\alpha)
    \begin{bmatrix}
    1 & 0 \
    0 & 0 \
    \end{bmatrix}
    R(-\alpha).
  \end{split}
\end{equation*}
$\alpha = 	an^{-1}(-\frac{1}{2})$, so we can get the matrix is 
\begin{equation*}
    T = 
    \begin{bmatrix}
    \frac{4}{5} & -\frac{2}{5} \
    -\frac{2}{5} & \frac{1}{5} \
    \end{bmatrix}.
\end{equation*}

Exercise 1.5.4

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Exercise 1.5.4

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