Exercise 1.5.1

Answers

If $u$ and $v$ are orthogonal unit vectors, then we have

{(u + v)}^{T} (u - v) = (u^{T} + v^{T}) (u - v) = u^{T} u - u^{T} v + v u^{T} - v^{T} v = 1 + 0 - 0 - 1 = 0,

so $u + v$ is orthogonal to $u - v$ .

Similarly,

| u + v |^{2} = {(u + v)}^{T} (u + v) = (u^{T} + v^{T}) (u + v) = u^{T} u + u^{T} v + v^{T} u + v^{T} v = 1 + 0 + 0 + 1 = 2,

so the length of vector $u + v$ is $\sqrt{2}$ .

Similarly,

| u - v |^{2} = {(u - v)}^{T} (u - v) = (u^{T} - v^{T}) (u - v) = u^{T} u - u^{T} v - v^{T} u + v^{T} v = 1 - 0 - 0 + 1 = 2,

so the length of vector $u - v$ is $\sqrt{2}$ .

niuers

2020-03-20 00:00