Linear Algebra and Its Applications · Problem 3.4.7

Orthogonality — Gilbert Strang

Mathematics · pythagoras-orthonormal-basis

Problem

Show, by forming

b^T b

directly, that Pythagoras's law holds for any combination

b = x_1 q_1 + \cdots + x_n q_n

of orthonormal vectors:

\|b\|^2 = x_1^2 + \cdots + x_n^2

. In matrix terms,

b = Qx

, so this again proves that lengths are preserved:

\|Qx\|^2 = \|x\|^2