# Proving subset of a Nullspace

1. Apr 29, 2014

### inter060708

1. The problem statement, all variables and given/known data

Given matrix A (size m x n), prove N(A) is subset of N( A^t A).

A^t is matrix A transposed.

2. Relevant equations

3. The attempt at a solution

My assumption is m < n, using definition of nullspace, I ended up with N( A^t A) = a set of zero vector, while N(A) is not entirely included in N( A^t A).

Thank You.

2. Apr 29, 2014

### jbunniii

If $Ax = 0$, then what is $A^t A x$?

3. Apr 29, 2014

### inter060708

A^t A x= 0

and how do I justify the subset part?

4. Apr 29, 2014

### jbunniii

If $X$ and $Y$ are sets, how do you prove that $X \subset Y$ in general?

5. Apr 29, 2014

### inter060708

I need to show that elements in X also belongs to Y.

6. Apr 29, 2014

### jbunniii

Yes, that's right. So what is the defining property of an element of $N(A)$? In other words, $x \in N(A)$ if and only if ...?

7. Apr 29, 2014

### inter060708

x ε N(A) iff Ax = 0 and since A^t A x = 0 then x ε N(A^t A).
Therefore x belongs to both N(A) and N(A^t A).

Is this correct?

8. Apr 29, 2014

### jbunniii

You have the right idea, but you need to state the logic correctly. The goal is to prove that if $x \in N(A)$ then $x \in N(A^t A)$.

So, suppose $x \in N(A)$. Then by definition, $Ax = 0$. Therefore...?

9. Apr 29, 2014

### LCKurtz

Do you understand why that is zero? You stated it but didn't prove it.

10. Apr 29, 2014

### inter060708

Ok I think I got it.

to prove that if $x \in N(A)$ then $x \in N(A^t A)$.

$x \in N(A)$. By definition, $Ax = 0$
therefore $A^t A x = 0$ which means $x \in N(A^t A)$ as well.

Therefore $N(A) \subset N(A^t A)$.

Thanks a lot jbunniii.

Last edited: Apr 29, 2014
11. Apr 29, 2014

Looks good!