Thread: Rank A and Basis of Rn

Hi guys, I would really apprieciate it if I could get some tips to proving this:

If A is an m x n matrix with columns c1, c2, ... cn, and rank A = n, show that (I'll let B = A-transpose here)), {Bc1, Bc2, ... Bcn} is a basis of Rn.

So far, I know that in order to show that it's a basis of Rn, it must span and be linearly independent.

I know rank A = rank B = dim(col A) = dim(row A) = n

I know that the columns of A are independent, therefore the rows of B are independent.
The product of BA is then independent because it is invertible, and is the matrix [Bc1 Bc2 ... Bcn].
I know the rows of A span Rn and therefore the columns of B span Rn.

Am I wrong with assuming BA is independent due to invertibility? Since I think I will have to use this in the proof. And can I get a nudge towards the next step?

Any help would be lovely, thank you!

Hey Kyo.

You can use the fact directly that det(A) <> 0 (and A is square) shows that all vectors are linearly independent and since det(A) = det(A^t) then A^t also has the same property.

I'm not not sure how they expect you to assert det(A) = 0, but once you do that the rest follows.

Thanks chiro, but are you referring to AB being square? My A matrix is actually an m x n matrix.

Whatever matrix you are using as your basis (or trying to prove that a basis exists).