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## Homework Statement

A is matrix m*n

show that nullspace of A is the subset of nullspace of A^T*A

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- Thread starter iamzzz
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- #1

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A is matrix m*n

show that nullspace of A is the subset of nullspace of A^T*A

- #2

HallsofIvy

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

- #3

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

- #4

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You mean, you guess that AI know Ax=0 and i guess A^T*A is also 0

- #5

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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 ...

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- #6

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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.

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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- #7

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i love you guys thanks

- #8

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i love you guys thanks

Yeah, all the babes dig the math nerds ;)

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