Exercise 2.3.8. Describe the column space of the matrix
and give a basis for it. Do the same for .
Answer:The second column of is twice the first column, so that the two vectors are linearly dependent. The column space consists of any vector of the form
where
is any real number; geometrically the column space is a line passing through the origin and the point
. The vector
serves as a basis for the space.
We have
Again we have the second column equal to twice the first, so the two vectors are linearly dependent. Also, we have so that the first column of
is a linear combination of the first column of
. The matrix
therefore has the same column space as
and the vector
can serve as its basis.
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
.