Nullspace of matrix A and the nullspace of A^T*A

[iamzzz]

Homework Statement

A is matrix m*n
show that nullspace of A is the subset of nullspace of A^T*A

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Have you tried anything at all? The standard way to prove "X is a subset of Y" is start "If x is in X" and use the properties of X and Y to conclude "then x is in Y".

So suppose x is in the null space of A. What can you say about $A^T Ax$?

 

[iamzzz]

I know Ax=0 and i guess A^T*A is also 0

 

[haruspex]

I know Ax=0 and i guess A^T*A is also 0

You mean, you guess that A^T*Ax is also 0, right? But why do you need to guess? If Ax = 0, you can easily prove A^T*Ax = 0.

 

[iamzzz]

You mean, you guess that A^T*Ax is also 0, right? But why do you need to guess? If Ax = 0, you can easily prove A^T*Ax = 0.

Done prove ?
So x is also in the nullspace of A^TA so done ? serioulsy ...

 

[Cbray]

The set of all values of x such that Ax=0 is the Null space of A.
The set of all values of x such that (A^(T)*A)x=0 is the Null space of A^(T)*A
Since A^(T)*A is linear, (A^(T)*A)x = A^(T)*(Ax)
Therefore A^(T)*(Ax)=0
A^(T)*(0) = 0 - For all values of x such that Ax = 0 - Therefore the set of solutions to Ax=0 is a subset of A^(T)*A

Now just turn that into mathematical notation.

 

[iamzzz]

i love you guys thanks

 

[Cbray]

i love you guys thanks

Yeah, all the babes dig the math nerds ;)