The Spectral Theorem

[SNOOTCHIEBOOCHEE]

1. The problem statement, all variables and given/known data

Prove that for any square matrix A, ker A= (im A*)\bot

2. Relevant equations

A* = A conjugate transpose

3. The attempt at a solution

so we want to show that all A that satisfies Ax=0 for some x, that set is equal to (im A*)\bot

But i really dont know what image of this could possibly be... we arent given a function.

[jimmypoopins]

But i really dont know what image of this could possibly be... we arent given a function.

it holds for any function.

prove it for any function (you should be able to)

[SNOOTCHIEBOOCHEE]

So the image of the null space of A* is equal to the null space of A.

Im thinking this proof leads me in the direction that im A* = A. then we are done. but i dont know how to show that.