# Column and Null space proof!

• Nov 27th 2009, 03:55 PM
joe909
Column and Null space proof!
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!
• Nov 27th 2009, 07:14 PM
HallsofIvy
Quote:

Originally Posted by joe909
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\$
\$\displaystyle Ax=0
\$

If presume you mean to say that if ABv= A(Bv)= 0, then, letting x= Bv, Ax= 0. Then you can stop there! If v is any r-dimensional vector, then ABv= 0 so that A(Bv)= 0. What ever Bv is, A(Bv)= 0. But the "column space" of B is simply the space of all vectors of the form Bv- the "image" of B. And the "null space of A" is simply the space of all vectors x such that Ax= 0. The fact that A(Bv)= 0 for all v says that every vector of the form Bv, that is, every vector in the column space of B, is in the null space of A.

Quote:

[FONT=Fn]Where the null space is the span of the set of all vectors x that satisy the equation.

What equation?

Quote:

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\$)
Quote:

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!

Talking about basis vectors is completely unnecessary.