Vector Spaces And Subspaces 2

Question 1

What is an example of a basis being infinite

Answer 1

An example of a basis being infinite is:

List 1,X,…,X^n is a basis of F[X]

Question 2

When is a list v1,…vN a basis of V

Answer 2

List v1,…,vn is a basis when every vector of vector space can be written as a linear combination of v1,…,vn

Question 3

What is the coordinate vector L of v with respect to basis alpha and what does this depend on

Answer 3

Coordinate vector L of v with respect to basis alpha is v=LiVi
This depends on order of vectors (unlike list being spanning/L.I/basis)

Question 4

When is a single vector independent

Answer 4

A single vector is independent when it isn’t the zero vector

Question 5

When is a list of vectors alpha spanning

Answer 5

A list of vectors alpha is spanning iff Ax=b has a solution for any b in the field where A is m x n matrix with colj(A) = Vj which happens iff REF of A has no zero rows

Question 6

When is a list of vectors alpha L.I

Answer 6

A list of vectors alpha is L.I when all columns in REF of A have a pivot. When solution to Ax=0 is x = 0(#columns <= #rows)

Question 7

Given list alpha of n vectors and Beta of m vectors of a vector space, when is n<=m

Answer 7

N<=m is when alpha is L.I and B is spanning

Question 8

Given list alpha of n vectors and Beta of m vectors of a vector space, when does n=m

Answer 8

N=m when both alpha and beta are bases

Question 9

When is a vector space V finite dimensional

Answer 9

Vector space V is finite dimensional if it has a finite basis and is infinite dimensional otherwise

Question 10

What is dim V when V is finite dimensional and example

Answer 10

Dim V is the number of vectors in basis V

E.g dim F^n = n

Question 11

When is dim V = infinity

Answer 11

Dim V = infinity when V is infinite dimensional

Question 12

A linear map phi V to W is injective iff …

Answer 12

A linear map phi is injective iff ker(phi) = 0, consequently phi is an isomorphism iff “” and image = W