Week 3 - Linear Independence, Vector Spaces and Subspaces, and Bases

Question 1

What is a vector space, V? List two properties of these vectors.

Answer 1

It is a set whose elements are called vectors (e.g. v1… vr).

So, if v, w ∈ V and c ∈ ℝ, we can form v + w ∈ V and cv ∈ ℝ. There should be a vector 0 ∈ V.

Question 2

What is linear dependence?

Answer 2

If there is a non-zero solution to x1v1 + … + xrvr = 0, then {v1 … vr} is linearly dependent because one of the vectors can be written as a linear combination of the others. Therefore, this implies that there is more than one solution.

Question 3

What is linear independence?

Answer 3

If the only solution is the trivial solution for x1v1 + … + xrvr = 0, then {v1 … vr} is linearly independent.

Therefore: {x1 … xr} = 0.

In other words, the number of pivots is the same as the number of columns (r = n).

Question 4

Suppose that:
v1…vr ∈ ℝn with r > n.
x1v1 + … + xrvr = 0 if Ax = 0, where vector v are the columns of A and x = x1…,xr.

Is matrix A linearly independent or dependent? Why?

Answer 4

A is an n x r matrix - it has more columns than rows (r > n).

The RREF of A can have at most n pivots (one per row), so there must be a free variable –> non-trivial solution (LD).

Therefore, any set of more than n vector in ℝn is linearly dependent.

Question 5

What is a subspace? What are the three conditions that need to be met?

Answer 5

A subspace is a subset W of ℝn, and it is a vector space.

Conditions:
(i) it is non-empty -> at least one vector in W, check it contains the zero vector.
(ii) if v, w ∈ W, then v + w ∈ W -> closed under addition.
(iii) if v ∈ W, c ∈ ℝn, then cv ∈ W -> closed under scalar multiplication.

Question 6

When is a set of vectors (S) a basis?

Answer 6

We say that S is a basis if it is both linearly independent and a spanning set, if the equation x1 + … + xrvr = b has exactly one solution for any b ∈ V.

A is a m x n matrix, and r is the number of pivots.
S is LI (r = n)
span S = ℝm (r = m)

Therefore, a basis is (r = m = n).

Question 7

What does a basis provide a notion for?

Answer 7

A basis provides a notion of a coordinate system, whose shape is specified by its dimensions.

Question 8

If a set of vectors is a spanning set, what does it tell us?

Answer 8

For the equation x1 + … + xrvr = b, there is at least one solution for any b ∈ V.

Question 9

If a set of vectors is linearly independent, what does it tell us?

Answer 9

For the equation x1 + … + xrvr = b, there is at most one solution for any b ∈ V.

Question 10

List three types of subspaces in ℝ2.

Answer 10

{0}
ℝ2
For any non-zero v, the set {span{v} or cv | c ∈ ℝ}

Question 11

List four types of subspaces in ℝ3.

Answer 11

{0}
ℝ3
A line through 0 = span{v}
A plane through 0 = span{v,w}