Transpose 2/bilinear Maps

Question 1

What is dim sol (im(phi^T))

Answer 1

Dim sol (im(phi^T)) = dim V - dim im(phi^T)

Question 2

What is rank (phi)

Answer 2

Rank (phi) = rank (phi^T)

Question 3

When does linear map phi inject and surject

Answer 3

Linear map phi injects iff phi^T surjects and surjects iff phi^T injects opposite properties

Question 4

When is a map B bilnear and what is B(u,0)

Answer 4

Map B is Bilinear if it’s linear in each slot, i.e
B(Lu1 + u2, v) = LB(u1,v) + B(u2,v)
B(u, Lv1 + v2) = LB(u,v1) + B(u,v2)
B(u,0) = B(u, 0+0) = 0 = B(0,v)

Question 5

What is a bilnear pairing

Answer 5

Bilinear pairing is a Bilinear map B: U x V to F

Question 6

What is a Bilinear form

Answer 6

Bilinear form is B: U x U to F

Question 7

What are examples of Bilinear maps

Answer 7

Examples of Bilinear maps are:
Matrix multiplication
Composition of linear maps
Real I.P

Question 8

What is Bil(U,V ; W)

Answer 8

Bil(U,V ; W) is set of Bilinear maps and is a vector space

Question 9

What is matrix A representing Bilinear form B

Answer 9

Matrix A representing Bilinear form B is
Aij = B(Vi,Vj) where Vi,Vj are basis vectors of V

Question 10

What is A’ when B has matrices A, A’ w.r.t bases Beta and B’

Answer 10

When B has matrices A,A’ w.r.t bases beta and B’
A’ = P^TAP where P is change of basis matrix from Beta to B’