7

Question 1

For x, y ∈ Rⁿ, the inner product (sometimes called the dot product or scalar product) is
dened to be the number <x, y> given by

Answer 1

<x, y> = x^Ty = x₁y₁ + x₂y₂ + … + x_ny_n

Nb, The inner product is just a number, not another vector or
a matrix.

Question 2

An inner product on V is a mapping from (or operation on) pairs of vectors x, y to the real
numbers, the result of which is a real number denoted

<x, y>, which satises the following properties:

Answer 2

(i) <x, x> ≥ 0 for all x ∈ V , and <x, x> = 0 if and only if

• *x** = 0, the zero vector of the vector space
  (ii) <x, y> = <y, x> for all x, y ∈ V
  (iii) <αx + βy, z> = α<x, z> + β<y, z> for all x, y, z ∈ V and all α,β ∈R.

Question 3

An inner product space is when

Answer 3

vector space has an inner product defined on it

Question 4

The norm or length IIxII of a vector x is

Answer 4

IIxII = √<x,x> = √(x₁² + x₂² + x_n²)

This is the standard
Euclidean length of a vector

<x,x> = IIxII²

Question 5

x and y are
orthogonal if

Answer 5

x^Ty = 0

<x,y> = 0

x⟂y means x and y are orthogonal

Question 6

If a set of (non-zero) vectors are pairwise orthogonal

(that is, any two are orthogonal)
then it turns out that the vectors are linearly

Answer 6

independent

Question 7

{v₁, v₂, …, v_k} is a linearly independent set of vectors if

Answer 7

V is an inner product space and vectors
{v₁, v₂, …, v_k} ∈ V are pairwise orthogonal,

i.e. v_i⟂v_j where i≠j … so <v_i, v_j> = 0

Question 8

A matrix P is
orthogonal if

Answer 8

P^T = P^-1. Now, this means that P^TP = I (which is the condition that means the vectors are length 1 and vs are pairwise orthogonal (v^Tv =0)

A matrix A is orthogonally diagonsable if there is a

P^T = P^-1 and P^TAP = D

Question 9

A matrix P is orthogonal if and only if the columns of P form an
orthonormal set of vectors, meaning

Answer 9

When a set of vectors {x₁, x₂,…, x_k} is such that any two are orthogonal and,
furthermore, each has length 1 (IIx_iII² = 1, and so IIx_iII = 1

Question 10

Cauchy-Schwarz inequality

Answer 10

Suppose that V is an inner product space.

Then I<x, y>I ≤ IIxII IIyII
for all x, y ∈ V

Question 11

Generalised Pythagoras Theorem

Answer 11

IIx + yII² = IIxII² + IIyII²

Question 12

Triangle inequality for norms:

In an inner product space V , if
x, y ∈ V, then

Answer 12

IIx + yII ≤ IIxII + IIyII

Question 13

The Gram-Schmidt
orthonormalisation process is a way of

Answer 13

producing k vectors that span the same space as
is spanned by {v₁, v₂,…, v_k}, and that form an orthonormal set.

That is, the process produces a set {e₁, e₂,…, e_k} such that:

• Lin {e₁, e₂,…, e_k} = Lin {v₁, v₂,…, v_k}
• {e₁, e₂,…, e_k} is an orthonormal set.

Question 14

Gram-Schmidt
orthonormalisation process steps:

Answer 14

1. Set e₁ = v₁ / IIv₁II
2. Define u₂ = v₂ - <v₂,e₁>e₁

_{3. Set} e₂ = u₂ / IIu₂II

4. Define u₃ = v₃ - <v₃,e₂>e₂ - <v₃,e₁>e₁

_{5. Set} e₃ = u₃ / IIu₃II

6. State resulting set {e₁, e₂,…, e_k} is such that:

• Lin {e1, e2,…, ek} = Lin {v1, v2,…, vk}
• {e₁, e₂,…, e_k} is an orthonormal set.

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