## Linear Algebra and Its Applications, Exercise 3.2.12

Exercise 3.2.12. Find a projection matrix $P$ that projects every vector in $\mathbb{R}^2$ onto the line described by the equation $x+2y=0$.

Answer: One solution to the equation $x+2y=0$ is $a = \left( -2, 1 \right)$. The projection matrix that projects vectors onto the line through $a$ is

$P = aa^T/a^Ta = \frac{1}{-2 \cdot -2 + 1 \cdot 1} \begin{bmatrix} -2 \\ 1 \end{bmatrix}\begin{bmatrix} -2&1 \end{bmatrix}$

$= \frac{1}{5} \begin{bmatrix} 4&-2 \\ -2&1 \end{bmatrix} = \begin{bmatrix} \frac{4}{5}&-\frac{2}{5} \\ -\frac{2}{5}&\frac{1}{5} \end{bmatrix}$

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

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