Maximal L.I Lists And Minimal Spanning Lists

Question 1

What does it mean if alpha is a basis

Answer 1

If alpha is a basis, then it means that alpha is a maximal L.I list and a minimal spanning list

Question 2

What is a Maximal L.I list

Answer 2

Maximal L.I list is a list alpha such that there is no list B > = alpha and is L.I

Question 3

What is a minimal spanning list

Answer 3

A minimal spanning list is a list alpha such that there is no list B < = alpha and B is spanning

Question 4

What does it mean if W < = V (W is a subspace of V)

Answer 4

If W < = V then it means dim W < = dim V

Question 5

What does any finite spanning list in V contain

Answer 5

Any finite spanning list in V contains a basis

Question 6

What can any L.I list be extended to make

Answer 6

Any L.I list can be extended to a basis of vector space (if V is finite dimensional)

Question 7

What is the sifting algorithm for constructing bases

Answer 7

Sifting algorithm for constructing bases is:
Omit every vector such that Vj = span(V1,…,Vj-1) and list satisfies the following:
Sifted list is independent
If Vk+1 =/ span(V1,…,Vk) then keep it and fully sifted list is L.I and span = V so is a basis

Question 8

When does a matrix A represent a linear map phi

Answer 8

Matrix A represents linear map phi when:
Phi(Vj) = sum Aij Wi for all j
Matrix is unique as this is w.r.t bases alpha beta

Question 9

What is a linear operator

Answer 9

Linear operator is a linear map with domain = codomain