Here are my problems:

Prove that if C is am mxn matrix over the field R of reals then R^n-Col(C^T) (direct sum) N(C), where Col (A) denotes the column space of the matrix A and N(A) denotes the right null space of A.

Plese, help!

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- Aug 30th 2008, 05:58 PMmivanovacolumn space, null space
Here are my problems:

Prove that if C is am mxn matrix over the field R of reals then R^n-Col(C^T) (direct sum) N(C), where Col (A) denotes the column space of the matrix A and N(A) denotes the right null space of A.

Plese, help! - Aug 30th 2008, 06:11 PMThePerfectHacker
Note sure what the question is asking? Is that that this expression is zero?

If thus, I think you need to use the rank nullity theorem.

Define as a linear transformation .

But is the dimension of the column space.

And is the dimension of the null space.

Thus the dimensions add up to , the dimension of . - Aug 31st 2008, 10:49 AMmivanovacolumn space, null space
Hi,

Thank you so much for your answer. I don't think that this expressiin is zero. The way that is giving to us is to prove that R^n=Col(C^T)(dir sum)N(C).

And, sorry if it is a stupid questions, but what differrence will it make if it is "right" null space or "left"?

Thank you so much!