## 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 .

This entry was posted in linear algebra and tagged . Bookmark the permalink.