Describe the set of vectors orthogonal to 1 0 -3t show that

Question

1. Prove the Cauchy-Schwarz inequality, proposition 3.5, which states that ∀v, w ∈ V , where V is a Euclidean space then |(v|w)| ≤ ||v||. ||w||. (5)

Tip: Consider the properties of ||v + λw||² ≥ 0 as a polynomial in λ.

2. Starting from proposition 3.6, the triangle inequality, derive the following alternative forms of it:

||v -w|| ≤ ||v|| + ||w|| , ||v + w|| ≥ ||v|| -||w|| , and ||v -w|| ≥ ||v|| -||w||,

for v, w ∈ V , where V is a Euclidean space. (4)

3. Prove that in any complex inner product space (4)

(a) (u|v) = 1/2||u + v||² + i||u - iv||² - (1 + i)(||u||² + ||v||² ), and

(b) (u|v) = 1/4||u + v||² - ||u - v||² + i||u - iv||² - i||u + iv||².

4. Determine whether

(u|v) = (u₁ - v₁)² + (u2 - v₂)²

is an inner product on R₂ with u = (u₁, u₂)T , and v = (v₁, v₂)T . (2)

5. Let V = R³ with the standard inner product. Describe the set of vectors orthogonal to (1, 0, -3)^T. Show that this set is a subspace of V, find a basis and hence determine the dimension of the subspace. (5)