Duality 2

Question 1

What is dim annu U

Answer 1

Dim ann U = dim V - dim U

Question 2

What are properties of annihilators U,W <= V

Answer 2

Properties of annihilators are:
U <= W implies Ann W <= ann U
Ann(U+W) = ann U N annW
Ann U + ann W <= ann (U N W) with equality when V is finite dimensional

Question 3

What do ann and sol do

Answer 3

Ann and sol reverse inclusions and swap sums and intersections

Question 4

When is U <= sol E

Answer 4

U <= sol E iff E <= ann U , U <= V and E <= V*

Question 5

What is relationship between Sol and ann

Answer 5

Relationship between sol and ann is:
U <= sol(ann U)
E <= ann(sol E)
Mutually inverse bijection for V finite dimensional
U <= V and E <= V*

Question 6

How can we view sol and ann

Answer 6

Can view sol and ann as maps.
Ann : subspace of V to subspace of V*
Sol: subspace of V* to subspace of V

Question 7

What is transpose of linear map phi
Is this linear

Answer 7

Transpose of linear map phi is:
Map phi^T: W* to V* given by phi^T(alpha) = alpha o phi for all alpha in W*
This is linear map

Question 8

What are examples of transpose

Answer 8

Examples of transpose are:
Id V^T = idV*
(Psi o phi)^T = phi^T o psi^T

Question 9

What is matrix representing phi^T if matrix representing phi is A

Answer 9

Matrix representing phi^T = A^T

Question 10

What are ker(phi) and ker(phi^T)

Answer 10

Ker(phi) is sol (im(phi^T)) and ker(phi^T) = ann(im(phi))

Question 11

What are im(phi) and im(phi^T)

Answer 11

Im(phi) <= sol (ker(phi^T)) and im(phi^T) <= ann(ker(phi)) with equality if V,W finite dimensional