Let a be a 6 × n matrix of rank r and let b be a vector in r6. for each choice of r and n that follows, indicate the possibilities as to the number of solutions one could have for the linear system ax = b. explain your answers. (a) n = 7, r = 5 (b) n = 7, r = 6 (c) n = 5, r = 5 (d) n = 5, r = 4

First, recall the rank-nullity theorem, which says for an matrix , the following holds:



That is, the rank and dimension of the nullspace of add up to the number of columns. Note that any time , we have at least one vector in the nullspace of , which can contribute to the system having infinitely many solutions.

a. If and , then



so there would be infinitely many solutions.

b. If and , we get 1 in place of 2 in part (a), so again, there would be infinitely many solutions.

c. If , then we'd get a nullity of 0. This means that, in order for a solution to exist to , the columns of must be a linear combination of . Otherwise, there would be no solution to the system.

d. If and , then the nullity is 1, so there would be infinitely many solutions.

Indeed, any time , the nullity would be non-zero, meaning there will always be infinitely many solutions in this case.