Linear Algebra and Its Applications, Review Exercise 2.23

Review exercise 2.23. Given any three vectors v_1, v_2, and v_3 in \mathbb{R}^3 find a matrix A that transforms the three elementary vectors e_1, e_2, and e_3 respectively into those three vectors.

Answer: When multiplying e_1 by A only the entries in the first column of A are multiplied by one; all other entries are multiplied by zero. So the first column of A should be set to v_1. Similarly when multiplying e_2 by A only the entries in the second column of A are multiplied by one; all other entries are multiplied by zero. The second column of  A should therefore be set to v_2. Finally the third column of A should be set to v_3.

If v_1 = (a_1, a_2, a_3), v_2 = (b_1, b_2, b_3), and v_3 = (c_1, c_2, c_3) then we have

A = \begin{bmatrix} a_1&b_1&c_1 \\ a_2&b_2&c_2 \\ a_3&b_3&c_3 \end{bmatrix}

so that (for example)

Ae_2 = \begin{bmatrix} a_1&b_1&c_1 \\ a_2&b_2&c_2 \\ a_3&b_3&c_3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}

= \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = v_2

Note that if A is invertible if and only if the three vectors v_1, v_2, and v_3 are linearly independent.

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 , , . Bookmark the permalink.

Leave a Reply

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

WordPress.com Logo

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

Twitter picture

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

Facebook photo

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

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s