Thread: Column and Null Space?

This was a question on my test and I couldn't figure it out.

Suppose A is m x n matrix, Prove the following equality:

dim Col(A) +dim Nul(A^T) = m

I had no clue how to do this. I know rank(A)=rank(A^T) but that's it.

do you know the rank nullity theorem?

What's that? Is it when If matrix A hasn columns, then rankA+ dimNulA = n?

that is one version of it, yes.

That's the only theorem we have. I don't know what else other than rankA=rank(A^T)

if A is an mxn matrix, then A^T is an nxm matrix.

apply the rank nullity theorem to A^T:

m = rank(A^T) + dim(null(A^T)).

now, what is your definition of rank(A)?