



Here is a more “modern” way to look at Pythagoras’ theorem. The statement a^2 + b^2 = c^2 is equivalent to the assertion that the matrices \begin{pmatrix} a & b \\ -b & a \end{pmatrix} and \begin{pmatrix} c & 0 \\ 0 & c \end{pmatrix} have the same determinant. But it is easy to see geometrically that the linear transformations associated to these matrices differ by a rotation, and the claim follows.

Homework: why are the above two proofs essentially the same proof?