Linear Operators On Inner Product Spaces

Question 1

What does non degeneracy lemma state

Answer 1

Non-degeneracy lemma states that:

Inner(v,w) = 0 for all w in V iff v =0

Question 2

What is adjoint of (phi o psi)* and (phi + Lpsi)

Answer 2

Adjoint of (phi o psi)* is psi* o psi* and 
(Phi + L*psi)* = phi* + L(bar)psi*

Question 3

When does phi have unique adjoint phi*

Answer 3

Phi has unique adjoint phi* when!

V is in an I.P space with an orthonormal basis and phi E L(V) (linear operators)

Question 4

When is a sq matrix A a hermitan or skew-Hermitan

Answer 4

Sq matrix A is a hermitan when:
A(dagger) (complex conjugate transpose) = A 
Skew-Hermitan when!
A(dagger) = -A
If field = C

Question 5

When is a sq matrix A symmetric or skew-symmetric

Answer 5

Sq matrix A is symmetric when:
A^T = A
Skew-symmetric when:
A^T = -A
When field = R

Question 6

What are properties of linear isometries

Answer 6

Properties of linear isometries are:
Norm-preserving norm(phi(v)) in W = norm(v) in V for all v in V
Distance preserving
Norm(phi(v1) - phi(v2)) in W = norm(v1 - v2) in V for all v1,v2 in V

Question 7

When is phi a linear isometry in V

Answer 7

Phi is a linear isometry in V iff phi^-1 = phi*