Pythagorean theorem

• Works in any dimension!
• Allows measuring the 2-norm of a vector by measuring the length of its components along orthogonal axes.
• Take $x=u+v$ where $u$ and $v$ are orthogonal. Then:

$$\Vert x{\Vert }_2^2=\Vert u{\Vert }_2^2+\Vert v{\Vert }_2^2$$

• Note that this is true for any decomposition of $x$ as long as $u$ and $v$ are orthogonal.

Vector norms