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

## Homework Statement

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

## The Attempt at a Solution

HallsofIvy
Homework Helper
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 $A^T Ax$?

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

haruspex
Homework Helper
Gold Member
2020 Award
I know Ax=0 and i guess A^T*A is also 0
You mean, you guess that AT*Ax is also 0, right? But why do you need to guess? If Ax = 0, you can easily prove AT*Ax = 0.

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

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

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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.

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