Basis and Dimension

Goals: Know the definitions for a basis for a vector space
and the dimension of a vector space.

I. The Span Game

For R²

We begin the Span game by choosing any two vectors u and v in R².

Let

and

(Try changing u and v.)

Next, you give me any vector z in R², and if I can find a c₁ and a c₂
such that I win. If not, you win.

No peeking Hank!

You give me:

I give you:

For R³

Let

You give me

For me to win, I must give you a C₁, C₂ and C₃ such that

I give you:

Can you beat me? How?

In the span game we simply find out if the vector z is an element (a linear combination) of the span of a given set of vectors. You may have noticed, that in order for you to win, my set of vectors u, v and w must be linearly dependent. The point of the span game is to help you gain some experience with spanning sets.

II. Basis

We know that the span of a set of vectors in a vector space V forms a subspace of V, but what if the span is all of V? Furthermore, what if the vectors in the spanning set are linearly independent? These conditions define one of the most important of all vector space properties, that of a basis.

Definition Basis: The vectors v₁, v₂, ... , v_n form a basis for a vector space V if and only if
i) v₁, v₂, ... , v_n are linearly independent
ii) The Span(v₁, v₂, ... , v_n ) = V

Basis vectors form the basic building blocks of any vector space, since every element in the vector space may be written as a unique linear combination of the basis vectors. Sometimes a basis is referred to as the minimal spanning set for V.

III. How to Check if a Set of Vectors is a Basis of V

Which of the following is a basis for R³?

a)

{e₁, e₂, e₃}

b)

{u, v, w} with

We need to check i) and ii).

Akamai: Bases for vector spaces are not unique!

Does the set {E₁₁, E₁₂, E₂₁, E₂₂} form a basis for R^2x2 , given

Solution: i) Check for Linear Independence

Let

Solve for the c_i's.

Are these four vectors Linearly Independent?

ii) Check the Span

Let

be any vector in R^2x2 .

Can we find c_i's such that ?

One more time (at least). Look for the similarity of method.

Does the set of vectors { 1, x, x², x³} form a basis for P₄?

Fill in the steps! Then do the calculation

i)

ii)

In your own words, tell how you test to see if a set of vectors forms a basis for a given vector space V.

Akamai: The concept of a Basis for Vector Space is one of the most important of all the structure properties. Be sure you understand this concept!

IV. How to Find a Basis for a Vector Space

Once we know how to test a given set of vectors for linear independence and find the span, the next goal is to be able to find a basis for a given vector space.

Find a basis for the nullspace of A, given

Solution: Since the Nullspace is defined as {x Î R³| Ax = 0}

which is equivalent to

We have two free variables to which we assign the scalar values

The nullspace of A is composed of vectors

Check this.

That is any x in the nullspace of A is a linear combination
of u = (2, 1, 0)^T and v = (-3, 0, 1)^T that is x Î Span(u,v).

Are u and v Linearly Independent?
Do these two vectors form a basis for Null(A)?

Do these two vectors span all of R³?

Find a basis for P₃.

Solution: By the Method of Construction.

(In this case we are free to choose!)

Since any vector in P₃ has the form

We need to find a set of linearly independent vectors v_i in P₃ such that z Î Span(v_i).

Choose a vector.

Does v₁ span P₃?
Can every vector in P₃ be written as a linear combination of v₁?

Since v₁ does not span P₃, we add another vector v₂.

i) Are v₁ and v₂ linearly independent?

ii) If so, do v₁ and v₂ span P₃?

Since v₁ and v₂ do not span P₃, we add a third vector. At this point you should be wondering how long we will need to add vectors before we get a spanning set.

Let

i) Are v₁ , v₂ and v₃ linearly independent?

ii) If so, do they span P₃?

Yes! therefore v₁ , v₂ and v₃ form a basis for P₃.

Therefore, every polynomial in P₃ may be written as a sum of v₁ , v₂ and v₃ .

V. Dimension

In the previous example, we needed three vectors to form a basis for P₃. If we chose some different set of vectors could we find a basis with only two vectors, or four vectors for P₃?

We know that there may be more than one basis for a given vector space, but do different bases have different numbers of elements? The answer to these questions is found in the next two theorems.

Theorem: If { v₁, v₂, ... , v_n} is a spanning set for a vector space V, then any collection of m vectors in V, where m > n is linearly dependent.

Proof: In class

So, by the Toss-out theorem we can whittle down the set of m vectors until we have n vectors left without changing the span of the original m vectors. So here is the next part of our answer.

Corollary: If { v₁, v₂, ... , v_n} and { u₁, u₂, ... , u_m} are both bases for a vector space V, then n = m.

Proof: In class

Akamai: Every basis for a given vector space has the same number of elements.

Definition of Dimension: The number n of elements in a basis for a vector space V is called the dimension of V. The vector space { O} has dimension 0. V is finite-dimensional if there are a finite number of vectors in any basis of V, otherwise V is infinte-dimensional.

What is the number of elements in any basis for Rⁿ?

What is the number of elements in any basis for P₃? For P_n?

So, if we have a set of n vectors in an n dimensional vector space, to check if they form a basis for V, all we need to show is...

Theorem: If V is a vector space of dimension n > 0:

I. Any set of n linearly independent vectors spans V.
II. Any n vectors that span V are linearly independent.

Proof: If we have time.

Show that

form a basis for R³.

Solution: Since the dimension 3 is the same as the number of vectors, we need to only check for Linear Independence.

Therefore u, v and w are linearly independent.

VI. Infinite Dimensional Vector Spaces

Show that C[a,b] is an infinite-dimesional vector space.

Solution: By construction. We need to show that we can find a infinte set of linearly independent vectors in C[a,b].

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Lecture 8 Notes