Ismorphisms

Question 1

When is a linear map an isomporphism?

Answer 1

If its inverse T(-1) : B -> A exists such that the composition between T and T(-1) creates the identity matrix of space A and the vice versa combination create id of space B

Question 2

When are two vector spaces isomporhic?

Answer 2

• If there exists isomporphism
• If they have the same dimensions

Question 3

• Let c be a column in a m-dimensional column VS
• Let V be space with basis B
  What is the vector in V corresponding to c wrt to the basis B?

Answer 3

c = [c1, c2, c3,…,cm]
then vec = c1 e1 + c2e2+…+cmem

Question 4

What is the inverse of the linear map that sends vectors to coordinate vectors ito B?

Answer 4

Colm -> V
or c -> vec(V, B) (c)

Question 5

Under which conditions is injectivity, surjectivity and isomorphism equivalent (for finite sets and for vector spaces)?

Answer 5

• Injectve if kernel = 0
• Surjective if image = codomain
• Bijective if injective and surjective

Equivalent when dimensions of domain and codomain are equal, T is surjective iff T is injective, T is injective iff T is surjective

Question 6

What does it mean when a LM is isomporphic?