Answer:
Proofs for Pythagoras Theorem usually use visual/geometry approaches. I don't post pictures in my answers, so I will present a linear algebra approach. You can see it in the blog posted by Professor Terence Tao.
Note that there are several elegant proofs using animations and drawings, but this is just personal.
I've seen this some time ago, it is really interesting proof.
It states that is equivalent to the statement that the matrices
and
have the same determinant.
The determinant of the first matrix is
The determinant of the second matrix is
Once the linear transformations associated with these matrices differ by rotation, we claim that