Exercise 2.4.5. Suppose that for two matrices and . Show that the column space is contained within the nullspace and that the row space is contained within the left nullspace .

Answer: Assume that is an by matrix and is an by matrix, and that are the columns of . Since each row of multiplies each column of to produce zero, we must have for all . In other words, each is in the nullspace of .

Let by any vector in the column space of . We then have

for some set of coefficients and thus

So any vector in the column space of is also in the nullspace of , and thus is contained within .

Similarly, let be the rows of . Since each row of multiplies each column of to produce zero, we must have for all . In other words, each is in the left nullspace of .

Let by any vector in the row space of . We then have

for some set of coefficients and thus

So any vector in the row space of is also in the left nullspace of , and thus is contained within .

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.