Quotients

Question 1

What is an affine subspace

Answer 1

Affine subspace is a subspace that doesn’t necessarily go through origin

Question 2

What is a coset

Answer 2

Coset is a set U = (v + u | u in U) <= V (coset of U in V) v is coset representative

Question 3

What is the fibre of a point

Answer 3

Fibre of a point is all points that map to it, fibre of w is all v in V s.t phi(v) = w

Question 4

What is quotient space V/U

Answer 4

Quotient space V/U is coset of U V/U = (v + U | v in V) subset of P(V)

Question 5

What are properties of quotient map q V to V/U

Answer 5

Properties of quotient map q V to V/U are:
Q surjects
Ker q = U

Question 6

What can we do with cosets

Answer 6

With cosets, we can add and scalar multiply them to make V/U into a vector space and q into linear map
(v + U) + (w + U) = (v+w) + U
L(v+U) = Lv + U

Question 7

If q: V to V/U, what are kernel and image

Answer 7

If q: V to V/U, kernel q = U and I’m q = V/U check fig 2.5

Question 8

What is dim(V/U)

Answer 8

Dim(V/U) = dim V - dim U

Question 9

What does 1 st isomorphism theorem state

Answer 9

1st isomorphism theorem states that :
If phi: V to W is a linear map then V/ker(phi) isomorphic to im(phi)
See diagram in lecture 8 at end