(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A is a mxn matrix, and P is an invertible nxn matrix.

So I want to prove that the bases of null(A) and null(AP) have the same number of elements.

2. Relevant equations

3. The attempt at a solution

I was going to start off by assuming that {X1, X2, ... Xm} is a basis of null(A). This is my first issue. Can I really just assume that?

So AXi = 0, following the definition of null space. Then V^{-1}Xi is in null(AV), since AV(V^{-1}Xi) = 0. Somehow I need to get from that to the fact that {V^{-1}X1, V^{-1}X2, ... V^{-1}Xm} is independent. But how?

I can prove that {V^{-1}X1, V^{-1}X2, ... V^{-1}Xm} spans null(A), I just need to know that it's independent for it to be a basis. And if it is, then it has m elements and so does null(A)'s basis.

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# Homework Help: Finding basis for null(A) and null(AP)

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