Review exercise 2.28. a) If is an by matrix with linearly independent rows, what is the rank of ? The column space of ? The left null space of ?
b) If is an 8 by 10 matrix and the nullspace of has dimension 2, show that the system has a solution for any .
Answer: a) If has rows and they are linearly independent then the rank of is .
Since the rank is there are also linearly independent columns of and the column space is . (In other words, the linearly independent columns span .)
The dimension of the left nullspace is then . The left nullspace thus contains only the zero vector.
b) Since the dimension of the nullspace of a matrix is in general, if the dimension of the nullspace of is 2 we have or . Since the rank of is equal to the number of rows of , by 20Q on page 96 there exists a solution to the system for every .
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.