I have a simple question.
Actually I just tried to solve the question 'Is range of $A$ equal to range of $AA^TA$'.
But it looks like much general question to ask 'Is range of $A$ equal to range of $AB$'.
In my first impression, I think range of $A$ is same as range of $AB$ because no matter which vectors come after, matrix $A$ would linearly transform it to the column space of $A$.
But, it looks wrong.
Can you help me to understand it?
And how can I prove range of $A$ is equal to $AA^TA$?
I am studying power iteration in randomized SVD and it said they are same but I cannot get it.
I guess something like '$Null(A)$ is equal to $Null(A^TA)$'would be helpful, but hard to apply it.
Thank you very much.