Chapter 6

Question 1

inner product (on V)

Answer 1

a function that takes each ordered pair (u,v) of elements of V to a number ⟨u, v⟩ in F and has the following properties:
- positivity: ⟨v, v⟩ ≥ 0 for all v in V
- definiteness: ⟨v, v⟩ = 0 iff v = 0
- additivity in first slot: ⟨u + v, w⟩ = ⟨u, w⟩ + ⟨v, w⟩ for all u, v, w in V
- homogeneity in first slot: ⟨λu, v⟩ = λ⟨u, v⟩ for all λ in F and all v in V
- conjugate symmetry: ⟨u, v⟩ = ˜⟨v, u⟩˜ (conjugate) for all u, v in V
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Question 2

Pythagorean Theorem

Answer 2

Suppose u and v are orthogonal vectors in V.

Then ||u + v||^2 = ||u||^2 + ||v||^2

Question 3

an orthogonal decomposition

Answer 3

Suppose u, v in V, with v ≠ 0.

Set c = ⟨u, v⟩ / ||v||^2
and w = u - (⟨u, v⟩ / ||v||^2)*v.

Then
⟨w, v⟩ = 0
and u = cv + w
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Question 4

Cauchy-Schwarz Inequality

Answer 4

Suppose u, v in V.
Then |⟨u, v⟩| ≤ ||u|| ||v||.
This inequality is an equality iff one of u, v is a scalar multiple of the other.

Question 5

Triangle Inequality

Answer 5

Suppose u, v in V. Then ||u + v|| ≤ ||u|| + ||v||. This inequality is an equality iff one of u, v is a nonnegative multiple of the other.

Question 6

orthonormal

Answer 6

• each vector in the list has norm 1 and is orthogonal to all other vectors in the list
• in other words, a list e1,…,em of vectors in V s.t. ⟨ej, ek⟩ = 1 if j = k, 0 if j ≠ k
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Question 7

orthonormal basis (of V)

Answer 7

an orthonormal list of vectors in V that is also a basis of V
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Question 8

Schur’s Theorem

Answer 8

Suppose V is a finite-dimensional complex vector space and T in L(V).
Then T has an upper-triangular matrix w.r.t. some orthonormal basis of V
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Question 9

Riesz Representation Theorem

Answer 9

suppose V is finite-dimensional and phi is a linear functional on V. Then there is a unique vector u in V s.t. phi(v) = ⟨v, u⟩ for every v in V

Question 10

How to write a vector as linear combination of orthonormal basis

Answer 10

Suppose e1,…,en is an orthonormal basis of V and v in V.
Then v = ⟨v, e1⟩e1 + … + ⟨v, en⟩en
and ||v||^2 = |⟨v, e1⟩|^2 + … + |⟨v, en⟩|^2

Question 11

What is the norm of an orthonormal linear combination?

Answer 11

if e1,…,em is an orthonormal list of vectors in V,

Then ||a1e1 + … + amem||^2 = |a1|^2 + … + |am|^2 for all a1,…,am in F

Question 12

Gram-Schmidt Procedure

Answer 12

Suppose v1,…,vm is a linearly independent list of vectors in V.

Let e1 = v1 / ||v1||.

For j = 2,…,m, define ej inductively by
ej = (vj - ⟨vj, e1⟩e1 - … - ⟨vj, e(j-1)⟩e(j-1)) /
||vj - ⟨vj, e1⟩e1 - … - ⟨vj, e(j-1)⟩e(j-1)||.

Then:
e1,…,em is an orthonormal list of vectors in V s.t. span(v1,…,vj) = span(e1,…,ej) for j = 1,…,m
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