Linear Algebra and Its Applications, Exercise 3.2.2 | My Math Homework

Linear Algebra and Its Applications, Exercise 3.2.2

Posted on April 26, 2014 by hecker

Exercise 3.2.2. Use the formula $p = \bar{x}a = \left(a^Tb/a^Ta\right)a$ (where $p$ is the projection of $b$ on $a$ ) to confirm that $\|p\| = \|b\||\cos \theta|$ (where $\theta$ is the angle between $a$ and $b$ ).

Answer: Since $p = \left(a^Tb/a^Ta\right)a$ we can compute the square of the length of $p$ as

$\|p\|^2 = \left[\left(a^Tb/a^Ta\right)a\right]^T \left[\left(a^Tb/a^Ta\right)a\right]$

Since $\left(a^Tb/a^Ta\right)$ is a scalar quantity we can bring it out of the transpose expression and then bring it to the left of the expression:

$\|p\|^2 = \left[\left(a^Tb/a^Ta\right)a\right]^T \left[\left(a^Tb/a^Ta\right)a\right]$

$= \left(a^Tb/a^Ta\right) a^T \left(a^Tb/a^Ta\right) a$

$= \left(a^Tb/a^Ta\right) \left(a^Tb/a^Ta\right) a^Ta$

$= \left(a^Tb/a^Ta\right)^2 a^Ta$

This leaves us with the scalar quantity $a^Ta$ at the right, so we can further simplify this expression as follows:

$\|p\|^2 = \left(a^Tb/a^Ta\right)^2 a^Ta = \left(a^Tb\right)^2 a^Ta / \left(a^Ta\right)^2$

$= \left(a^Tb\right)^2 / a^Ta = \left(a^Tb\right)^2/\|a\|^2$

Taking the positive square root of both sides we then have

$\|p\| = |a^Tb/\|a\||$

At the same time from (2) on page 146 (theorem 3G) we have

$\cos \theta = a^Tb/\left(\|a\|\|b\|\right)$

so that

$\|b\|\cos \theta = a^Tb/\|a\|$

Substituting into the above expression for $p$ we then have

$\|p\| = |a^Tb/\|a\|| = |\|b\|\cos \theta| = \|b\||\cos \theta|$

(since $\|b\|$ is guaranteed to be positive but $\cos \theta$ is not).

UPDATE: I expanded the derivation above in response to a question posted on MathHub.

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.

Buy me a snack to sponsor more posts like this!

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

1 Response to Linear Algebra and Its Applications, Exercise 3.2.2

Pingback: Question regarding trasposes and norms - MathHub

Leave a Reply Cancel reply

Enter your comment here...

Fill in your details below or click an icon to log in:

Email (required) (Address never made public)

Name (required)

Website

You are commenting using your WordPress.com account. ( Log Out / Change )

You are commenting using your Facebook account. ( Log Out / Change )

Connecting to %s

%d bloggers like this: