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
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.