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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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)?
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