Vector Spaces

Question 1

What is a vector space

Answer 1

A vector space is a set V which is an a abelian group and has scalar multiplication

Question 2

What are properties of an abelian group

Answer 2

Properties of an abelian group are:
V+w=w+v
U+(v+w)=(u+v)+w
0 is in group

Question 3

What are examples of vector spaces

Answer 3

Examples of vectors spaces are:
F^n
M x n matrices

Question 4

What is a vector/linear subspace

Answer 4

A vector/linear subspace of a vector space V is U <= V which is closed under addition and scalar multiplication. Must be Non-empty U is also now a vector space
U1, U2 in U, U1+U2 in U
U in U, LU in U

Question 5

What are examples of subspace

Answer 5

Examples of subspace are:
V itself

Question 6

When is U a subspace of V

Answer 6

U is a subspace of V when:
0 is in U
U1,U2 in U, U1 + LU2 in U

Question 7

What is an easy way to see if something is a vector space

Answer 7

Easy way to see if something is a vector space is to see it’s a subspace of some V^I

Question 8

What is span of a list of vectors

Answer 8

Span of a list of vectors is all linear combinations (subspace of V)

Question 9

When is a list of vectors spanning

Answer 9

A list of vectors is spanning when span(V1,..,VN) = V

Question 10

When are vectors L.I

Answer 10

Vectors are L.I when they can’t be written as linear combinations of each other