Exercise 3.3.3. Given
solve to find . Show that is orthogonal to every column in .
Answer: We have or if is invertible. In this case we have
We then have
The error vector is then
The inner product of the error vector with column 1 of is
The inner product of with column 2 of is
Thus is othogonal to all columns of (and thus to the column space of as well).
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.