Exercise 3.3.6. Given the matrix and vector defined as follows

find the projection of onto the column space of .

Decompose the vector into the sum of two orthogonal vectors and where is in the column space. Which subspace is in?

Answer: We have per equation (3) of 3L on page 156. We first compute

and then compute

Finally we compute

Since we have

We have

so that and are orthogonal.

The vector is in , the column space of A, and the orthogonal subspace of is , the left nullspace of . Since is in and is orthogonal to , must be in , so that . We confirm this:

UPDATE: I corrected the calculation of ; thanks go to KTL for pointing out the error.

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.

Thank you for the great work! It’s been really helpful. I wanted to mention that A transpose by A is [6 -8; -8 18] and not [9 -9; -9 18] that changes the results a bit.

I’m glad you find these posts useful. I’m sorry it’s been a long time since I published the last one.

I don’t understand your comment. Are you referring to A transpose multiplied by A (on the right)? If so, the (1, 1) element of the result matrix is 1 x 1 + 2 x 2 + (-2) x (-2) = 1 + 4 + 4 = 9. I don’t know how you got 6 for that result. Similarly the (1, 2) entry is 1 x 1 + 2 x (-1) + (-2) x 4 = 1 – 2 – 8 = -9 (not -8) and the (2, 1) entry is 1 x 1 + (-1) x 2 + 4 x (-2) = 1 – 2 – 8 = -9 again.

The question in the 4th edition is different. Hence the confusion for the person above 🙂

Ah, thanks for the explanation. I’ve never looked at a copy of the 4th edition so I don’t know how the exercises differ.

q you have given as p-b though the calculation is done as b-p (correctly)

Thanks for finding this error! I’ve corrected the post.