The problem M60, selected page 161, “A First Course in Linear Algebra”.
Suppose that {u, v, w} BELONGS TO C^n is an orthonormal set. Prove that u + v is not orthogonal to v + w.
library(knitr){u, v, w} are orthonormal.
Therefore, u.v = 0, v.w = 0 and u.w = 0. And u^2 = 1, v^2 = 1 and w^2 = 1.
Now, (u + v).(v + w) = u.v + u.w + v^2 + v.w = 0 + 0 + 1 + 0 = 1.
Therefore, (u + v).(v + w) = 1 != 0
Therefore, (u + v) is not orthogonal to (v + w).