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, 04:58 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, 05:11 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 $\displaystyle T(\bold{x}) = A\bold{x}$ as a linear transformation $\displaystyle A: \mathbb{R}^n \to \mathbb{R}^m$.

But $\displaystyle \text{rank}(A)$ is the dimension of the column space.

And $\displaystyle \text{nullity}(A)$ is the dimension of the null space.

Thus the dimensions add up to $\displaystyle n$, the dimension of $\displaystyle \mathbb{R}^n$. - Aug 31st 2008, 09:49 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!