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.

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- Nov 8th 2011, 07:19 PMReeferColumn 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. - Nov 8th 2011, 11:18 PMDevenoRe: Column and Null Space?
do you know the rank nullity theorem?

- Nov 9th 2011, 08:45 PMReeferRe: Column and Null Space?
What's that? Is it when If matrix A hasn columns, then rankA+ dimNulA = n?

- Nov 9th 2011, 09:00 PMDevenoRe: Column and Null Space?
that is one version of it, yes.

- Nov 9th 2011, 09:01 PMReeferRe: Column and Null Space?
That's the only theorem we have. I don't know what else other than rankA=rank(A^T)

- Nov 9th 2011, 09:07 PMDevenoRe: Column and Null Space?
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)?