Orthogonality

Question 1

What is triangle inequality

Answer 1

Triangle inequality is:

Mod(v+w) <= mod(v) + mod(w)

Question 2

What is parallelogram identity

Answer 2

Parallelogram identity is:

Mod(v+w)^2 + mod(v-w)^2 = 2(mod(v)^2 + mod(w)^2)

Question 3

When is a list of vectors U1,…,Uk in an inner product space orthonormal

Answer 3

a list of vectors U1,…,Uk in an inner product space is orthonormal when:
Inner product(Ui,Uj) = delta(ij) = 1 if i = j and 0 when i=/ j

Question 4

What is an orthonormal basis

Answer 4

Orthonormal basis is when a list of vectors U1,…Uk is orthonormal and also a basis

Question 5

What is a list of any orthonormal vectors

Answer 5

A list of any orthonormal vectors is linearly independent

Question 6

What is the inner product of 2 vectors in an inner product space V

Answer 6

Inner product of 2 vectors in inner product space V is:

Dot product of their coordinates

Question 7

What does Gram-Schmidt orthogonalisation state

Answer 7

Gram-Schmidt orthogonalisation states that:
If there is a list of V1,…,Vm L.I vectors in inner product space V:
Exists an orthonormal list u1,…,Um s.t
Span{u1,…,Uk} = span{v1,…,Vk} and more precisely
Uk = Wk/norm(Wk)
Where W1=V1

Question 8

What does Gram-Schmidt orthogonalisation state for k > 1

Answer 8

For k > 1, Gram-Schmidt orthogonalisation states that:
Wk = Vk - sum j =1 to k-1 inner(Wj, Vk)/norm(Wj^2) * Wj
And U1 = V1/norm(V1)
Where Uk = Wk/norm(Wk) (part of orthonormal basis)

Question 9

What does any finite dimensional inner product space have

Answer 9

Any finite dimensional inner product space has an orthonormal basis