# column space, null space

• Aug 30th 2008, 04:58 PM
mivanova
column 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 PM
ThePerfectHacker
Quote:

Originally Posted by mivanova
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.

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 AM
mivanova
column 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!