Exercise 2.3.19. Suppose is an by matrix, with columns taken from . What is the rank of if its column vectors are linearly independent? What is the rank of if its column vectors span ? What is the rank of if its column vectors are a basis for ?
Answer: If the column vectors of are linearly independent then there must be a pivot in every one of the columns, so that the rank .
If the columns of span then we must have . There can be no more than linearly independent vectors in so out of the columns of only columns can have pivots. Therefore the rank .
If the columns of are a basis for then they are linearly independent, which means the rank , and they also span so we must also have . We thus have .
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.