Linear Operators 2

Question 1

When is f distance preserving

Answer 1

F is distance preserving iff:
E Vo in V and a linear isometry phi s.t
F(v)= Vo + phi(v)

Question 2

When is subspace U of V phi invariant and example

Answer 2

Subspace U <= V is phi invariant if phi(u) <= U for all u in U
E.g set containing 0

Question 3

What is U perp

Answer 3

U perp is set s.t {v E V : inner(u,v) = 0 for all u in U}

Question 4

If U <= V is phi invariant, what does that mean for U perp

Answer 4

If U <= V is phi invariant, then U perp is phi* invariant

Question 5

When is linear operator phi normal

Answer 5

Linear map phi is normal if phi commutes with phi* (phi o phi* = phi* o phi)
E.g If phi is self adjoint or skew adjoint (phi* = -phi) then phi is normal

Question 6

If phi is Normal and U <= V is an eigenspace for phi, then what is U perp

Answer 6

If phi is Normal and U <= V is an eigenspace for phi, then U perp is phi invariant

Question 7

When is a linear operator phi orthogonally diagonalisable and what is phi

Answer 7

Linear operator phi is orthogonally diagonalisable if V has an orthonormal basis consisting of eigenvectors of phi.

Question 8

When does characteristic polynomial of phi have at least 1 root

Answer 8

Characteristic polynomial of phi (linear operator) has at least 1 root when field = C

Question 9

If phi is orthogonal diagonalisable, then what else is it

Answer 9

If phi is orthogonal diagonalisable then is it also normal and when field = R, phi is self adjoint

Question 10

If linear operators phi and psi commute and U is an eigenspace of phi, what is U

Answer 10

If linear operators phi and psi commute (psi o phi = phi o psi) and U is an eigenspace of phi then:
U is psi-invariant