Exercise 6.2.1

Answers

1.

No. It should be at least an independent set.

2.

Yes. See Theorem 6.5.

3.

Yes. Let

W

be a subspace. If

x

and

y

are elements in

W^{⊥}

and

c

is a scalar, we have

⟨ x + y, w ⟩ = ⟨ x, w ⟩ + ⟨ y, w ⟩ = 0

and

⟨ cx, w ⟩ = c ⟨ x, w ⟩ = 0

for all $w$ in $W$ . Furthermore, we also have $⟨ 0, w ⟩ = 0$ for all $w \in W$ .

4.

No. The basis should be orthonormal.

5.

Yes. See the definition of it.

6.

No. The set

{0}

is orthogonal but not independent.

7.

Yes. See the Corollary 2 after Theorem 6.3.

jephianlin

2011-06-27 00:00