Linear Algebra and Its Applications, Review Exercise 2.28

Review exercise 2.28. a) If A is an m by n matrix with linearly independent rows, what is the rank of A? The column space of A? The left null space of A?

b) If A is an 8 by 10 matrix and the nullspace of A has dimension 2, show that the system Ax = b has a solution for any b.

Answer: a) If A has m rows and they are linearly independent then the rank of A is r = m.

Since the rank is m there are also m linearly independent columns of A and the column space is \mathbb{R}^m. (In other words, the m linearly independent columns span \mathbb{R}^m.)

The dimension of the left nullspace is then m-r = m-m = 0. The left nullspace thus contains only the zero vector.

b) Since the dimension of the nullspace of a matrix is n-r in general, if the dimension of the nullspace of A is 2 we have 2 = 10-r or r = 8. Since the rank of A is equal to the number of rows of A, by 20Q on page 96 there exists a solution to the system Ax = b for every b.

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.

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