Exercise 6.2.15

Answers

1.

By Theorem 6.5, we have

x = \sum_{i = 1}^{n} ⟨ x, v_{i} ⟩ v_{i}, y = \sum_{j = 1}^{n} ⟨ y, v_{j} ⟩ v_{j} .

Thus we have

⟨ x, y ⟩ = \sum_{i, j} ⟨ ⟨ x, v_{i} ⟩ v_{i}, ⟨ y, v_{j} ⟩ v_{j} ⟩

= \sum_{i = 1}^{n} ⟨ ⟨ x, v_{i} ⟩ v_{i}, ⟨ y, v_{i} ⟩ v_{i} ⟩

= \sum {i = 1}^{n} ⟨ x, v_{i} ⟩ \bar{⟨ y, v_{i} ⟩} .

2.

The right hand side of the previous equality is the definition of the standard inner product of

𝔽^{n}

jephianlin

2011-06-27 00:00