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
so that
We then have
and
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
.