The question is :

Let

A be an m×n matrix and B an n×r matrix. If AB = O (the m×r matrix of zeroes)

prove that the column space of B is contained in the null space of A.

Here is what I have gotten:

We know:

$\displaystyle

AB=0

Ax=0

$

Where the null space is the span of the set of all vectors x that satisy the equation. Let null space(A) = span($\displaystyle x_1,x_2,...,x_s$)

I let col. space(B)=span($\displaystyle c_1,c_2,...,c_k$)

Let v be in the col. space(B)

Therefore $\displaystyle v=a_1c_1+a_2c_2+...+a_kc_k$ where $\displaystyle a_i $are elements of $\displaystyle R$

If we can show that v can be shown to be apart of the span($\displaystyle x_1,x_2,...,x_s$) then we are done. I cant think of a way to go about doing this.

Any help would be greatly appreciated!