Is N(A) a Subset of N(A^t A)?: Proving Inclusion for Matrix Nullspaces

[inter060708]

Homework Statement

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

A^t is matrix A transposed.

Homework Equations

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.

 

[jbunniii]

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

 

[inter060708]

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

A^t A x= 0

and how do I justify the subset part?

 

[jbunniii]

A^t A x= 0

and how do I justify the subset part?

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

 

[inter060708]

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

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

 

[jbunniii]

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

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 ...?

 

[inter060708]

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 ...?

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?

 

[jbunniii]

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?

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...?

 

[LCKurtz]

A^t A x= 0

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

 

[inter060708]

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...?

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.

 

[jbunniii]

Looks good!