## Related questions with answers

Find the matrix that projects every point in the plane onto the line x + 2y = 0.

Solution

VerifiedFirst, let's find a vector that belongs to the line $x+2y=0$. Since the equation implies that $x=-2y$, if we take $y=1$, we get that $a=(-2,1)^T$ belongs to the line $x+2y=0$. Now, let's compute the projection matrix $P$ onto the line through the vector $a$ (which is $x+2y=0$).

$\begin{align*} {\color{#4257b2}P}&=\dfrac{aa^T}{a^Ta}\\\\ &=\dfrac{\begin{bmatrix}-2\\\phantom{-}1\end{bmatrix}\begin{bmatrix}-2 & 1\end{bmatrix}}{\begin{bmatrix}-2 & 1\end{bmatrix}\begin{bmatrix}-2\\\phantom{-}1\end{bmatrix}}\\ &=\dfrac{\begin{bmatrix}\phantom{-}4 & -2\\-2 & \phantom{-}1\end{bmatrix}}{4+1}\\\\ &={\color{#4257b2}\begin{bmatrix}\phantom{-}4/5 & -2/5\\-2/5 & \phantom{-}1/5\end{bmatrix}} \end{align*}$

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