Linear Independence

Goals: To understand the vector space property of linear independence and how to determine if a given set of vectors are linearly independent. To be able find a minimal spanning set for a subspace.

We continue our with our explorations into the structure properties of Vector Spaces by trying to determine the subspaces within a given vector space. We have seen that every Vector Space must contain the zero vector subspace, O. But how do we find out if there are any other subspaces?

In the previous section, we observed that the span of any set of vectors {v_i} from a vector space V forms a subspace, the Span(v_i). You might wonder if it might be possible to use this idea in some way to find all the subspaces of V?
What role does the Span(v_i) play in this process? The key to spanning sets is that every vector in the span subspace can be written as a linear combination of the spanning vectors. By investigating linear combinations of the spanning vectors, we hope to find the minimal number of vectors needed to span a subspace. How do we intend to carry this out? We need to be able to determine if a given set of vectors are linearly independent.

I. Linear Independence

Definition Linear Independence: The vectors v₁, v₂,...,v_n in a vector space V are said to be Linearly Independent if c₁v₁ + c₂v₂ + ... +c_nv_n = 0 implies that all of the scalars c₁,...,c_n are equal to 0, otherwise the vectors are said to be Linearly Dependent.

Geometric Examples over R²

Linearly dependent vectors are easily illustrated in R² since linear combinations add up to 0, which means they form a loop.

The vectors u, v, and w are linearly dependent in R² since they add up to zero. In this example the scalars multiples of u, v and w are all 1.

Linearly Dependent

Linearly Independent

Geometric Examples over R³

Linearly Dependent

Linearly Independent

Now for some examples from our five Vector Spaces.

The Vector Space M^2x2

Check to see if the following vectors from M^2x2 are linearly independent.

Given

Notice that

implies that C is a linear combination of A and B, or C Î Span(A,B)

or since

we have nonzero coefficients yet the sum of these

is the zero vector.

Since we can write C as a linear combination of A and B, we say that these three vectors are linearly dependent.

However, if

We cannot write C as a linear combination of A and B. The only way that

or

Is if

Since we cannot write C as a linear combination of A and B, we say these three vectors are linearly independent.

Show me why the only solution to

is

Write out the sum as a singlke matrix.

this is equal to the zero matrix if and only if the following equations are satisfied.

since

we have

The Vector Space P₄

Given a set of vectors in P₄, how do we know if they are linearly independent?

Let

equating like terms:

Constant term

x term

x² term

x³ term

Are these vectors linearly independent?

or

Since all of the c_i are 0, these vectors are linearly independent.

The Vector Space R³

Are

linearly dependent in R³?

Solve

Keep in mind that we are interested in span subspaces, so we want to ask, What is the difference between the span of a set of linearly idependent vectors and the span of a set of linearly dependent vectors?

II. The Span of Linearly Independent Vectors

Find the span in R³ of

By definition the

Let's check these vectors for linear dependence.

We can view this as a homgeneous linear system of equations in Form 2.

This tells us that the vectors are linearly dependent. Why?

What we have then, is for u, v and w

What this indicates is that we can throw out the linearly dependent vectors and still have the same span, or

The Toss-out Theorem: Given n linearly dependent vectors v₁,...,v_n from a vector space V. If we toss-out vectors from v_n,...,v_k+1 until the remaining k vectors form a linearly independent set v₁,...,v_k, then
Span(v₁,...,v_n) = Span(v₁,...,v_k)
Proof: Exercise.

This theorem indicates a method for finding a minimal spanning set for a given finite vector space, one of our goals.

Another important property of linearly independent vectors is that every vector in the Span(v₁,...,v_n) is has a unique representation as a linear combination of the v₁,...,v_n. This is not true if the v₁,...,v_n are linearly dependent_.

Give an example over R³ of how a vector may be written in different ways as linear combinations of of a dependent set of vectors.

Let

then

and

Since we can write u in two different ways with respect to
v, e₁,e₂,e₃, these vectors are linearly dependent.

The Uniqueness Theorem: Let v₁,...,v_n be vectors in a vector space V. A vector v in the Span(v₁,...,v_n) can be written uniquely as a linear combination of v₁,...,v_n if and only if v₁,...,v_n are linearly independent.

Proof: (¬) If v Î Span(v₁,...,v_n), then v can be written as a linear combination
v = c₁v₁ + c₂v₂ + ... +c_nv_{n eq 1}
now suppose that v can also be written as a linear combination
v = a₁v₁ + a₂v₂ + ... +a_nv_{n eq2}
if v₁,...,v_n are linearly independent then subtracting eq1 from eq2 we must have

(a₁ - c₁)v₁ + (a₂ - c₂)v₂ + ... + (a_n - c_n)v_n = 0

Since the v₁,...,v_n are linearly independent
a₁ - c₁ = 0, a₂ - c₂ = 0, ..., a_n - c_n = 0
that is a₁ = c₁ a₂ = c₂, ..., a_n = c_n , thus

v = c₁v₁ + c₂v₂ + ... +c_nv_n is unique.
(®) Exercise

(Hint) Suppose c₁v₁ + c₂v₂ + ... +c_nv_n = 0 and not all of the c_i's are zero.

III. Testing for Linear Independence

The Toss-out theorem permits us to remove vectors from our spanning set until we obtain a set of linearly independent vectors without altering our span subspace. Now we look at some methods for determing whether a set of vectors are linearly independent.

The Vector Space Rⁿ

To determine whether or not a set of vectors is linearly independent in Rⁿ, we must solve a homogeneous system of linear equations, Ax = 0. Recall, if A is nonsingular, Ax = 0 has only the trivial solution.

Are the following vectors linearly independent over R³?

As before we look at the related system:

This can be put into the form Ax = 0.

with

Method 1: Find Det(A)

Since the Det(A) = 0, A is singular, and our vectors are linearly dependent.

Method 2: Find rref(A)

Given Ax = 0, and

What can you say about solutions of the related system?

Method 1, can be stated as a theorem.

Theorem: Let x₁, x₂, ..., x_n be n vectors in Rⁿ, if X is the matrix X = (x₁, x₂, ..., x_n ), then the vectors x₁, x₂, ..., x_n will be linearly dependent if and only if Det(X) = 0.

Proof: (Hint: Write the related linear system of equations)

State Method 2 as a theorem.

The Vector Space P_n

To test whether or not a set of polynomials p_i are linearly independent in Pⁿ, by definition we form:

c₁p₁ + c₂p₂ +c₃p₃+ ... +c_kp_k = O

By equating like terms, we obtain a homogeneous system of linear equations. Solving this system will indicate whether the polynomials are linearly independent.

Determine if the following polynomials are linearly independent.

we form:

grouping

Equating like terms

Conclusion?

The Vector Space C[a,b]

This vector space is a bit harder to test. Nevertheless if we are given a set of n functions, ( vectors in C[a,b]), and these functions are n-1 differentiable, we can use the following trick to test for linear independence if the functions are n-1 differentiable, C^(n-1).

The Wronskian

Definition The Wronskian:
Let f₁, f₂,...,f_n be n-1 differentiable functions in the vector space C[a,b]. Define
the function W[f₁, f₂, ... , f_n](x) on [a, b] by



.

Write out the Wronskian for the following sets of functions.

A)

B)

C)

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Lecture 7 Notes