Review exercise 2.30. Suppose that the matrix is a square matrix.

a) Show that the nullspace of contains the nullspace of .

b) Show that the column space of contains the column space of .

Answer: a) Suppose is in the nullspace of , so that . We then have so that is also in the nullspace of . Since this is true for every in the nullspace of , the nullspace of is a subset of the nullspace of or (stated differently) the nullspace of contains the nullspace of .

b) Suppose that is in the column space of . Then there exists some for which (in other words, is a linear combination of the columns of , with the entries of being the coefficients).

Let . Then we have . Since there exists a for which we see that is also in the column space of . Since this is true for every in the column space of , the column space of is a subset of the column space of or (stated differently) the column space of contains the column space 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.