Review exercise 2.24. Suppose that is a 3 by 5 matrix with the elementary vectors , , and in its column space. Does has a left inverse? A right inverse?
Answer: Since , , and are in the column space the dimension of the column space must be 3 (since , , and are linearly independent) and thus the rank of is , the number of rows of .
Since the rank of equals the number of rows has a 5 by 3 right inverse . However it does not have a left inverse since the rank is less than the number of columns .
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.