Exercise 3.3.8. Suppose that is a projection matrix from
onto a subspace
with dimension
. What is the column space of
? What is its rank?
Answer: Suppose that is a arbitrary vector in
. From the definition of
we know that
is a vector in
. But
is a linear combination of the columns of
, so that
also is in the column space
. Since any vector in
can be expressed as
for some
, all vectors in
are also in
, so that
.
Now suppose that is an arbitrary vector in the column space
. Then
can be expressed as a linear combination of the columns of
for some set of coefficients
. Consider the vector
. We then have
by the definition of
. But if
for some
then
is in
. So all vectors in
are also in
and thus
.
Since and
we then have
: The column space of
is
.
The rank of is the dimension of its column space. But since
is the column space of
, the rank of
is
, the dimension of
.
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
.