On Linear Independence ______(1) a Finite Set of Vectors That Contains a Zero Vector Is Linearly Dependent

Linear Independence, Spans, and Bases

Linear Independence

A set of vectors is linearly independent if

Otherwise it is linearly dependent. is linearly dependent means that there are vectors in which can be represented as linear combination of the other vectors

For example:

are linearly independent

Can we find a,b,c such that

Notice

Det(

The system has only one solution and that one is a=0,b=0,c=0.

For example:

are linearly dependent(Keary)

Can we find a,b,c such that

Solution: a=-2, b=1,c=1

Yes. This set of vectors is linearly dependent.

The minimal number of independent vectors in the subspaces is called the rank or dimension of the subspace.

How do we find the dimensions of the four spaces? Answer: Gauss-Jordan Elimination process

The first 3 columns are independent. Why?

This means we have only one, the trivial solution.

Because the system has only one solution

The fourth one is linearly dependent of the others. How about rows?

then ... What else could happen?

The Rank is 2.

The Rank is 1.

Notice: Rank(Col(A))=Rank(Row(A)) ______Let

The rank of the Null(A) is called nullity. Find the Rank and Nullity

Notice: Rank + Nullity=the number of Columns The Rank and Nullity Theorem

More on Linear Independence ______(1) A finite set of vectors that contains a zero vector is linearly dependent

Let

Is S linearly independent or dependent? Dependent.

______(2) A set of two vectors is linearly independent iff neither of the vectors is a multiple of the other

Let

Is S linearly independent or dependent? Independent, because the two vectors are not multiple of each other. ______

(2) Let be a set of vectors in . If , then S is linearly dependent.

S is dependent. ______

Let

be a set of vectors Then

i.e. Span(S) is a set of all vectors which can be represented as a linear combination of vectors in S. Span(S) is a vector space. Example: Let

A Basis of the Vector space

If V is a vector space and is a set of vectors in V. Then S is called a basis if (1) S is linearly independent (2) span(S)=V Example: canonical example, V= Canonical Basis

Pick .

John’s

Give me another basis for

Q:Is T a basis? Check independence! Det=1 hence they are linearly independent.

John’s

Does this system have a solution? Det=1 means it has the solution no matter what u you pick. T is a basis for

Theorem: A vector representation in a basis is unique. Proof: Let u have two representation in the same basis.

and

Then

Since s are independent this can happen only if

Therefore

and representation is unique. Theorem. Every vector space has a basis. All bases of the same vector space have the same number of elements. This number is called a DIMENSION of the vector space. Example and Problems: Let

Q: Is in span(S)?

Yes, because

Chad says: It is not in the span!!!!!

______

Is this set linearly independent?

We performed Gauss –Jordan on columns and we learned the following: S is linearly independent. S is a basis for its span. Also

T is also a basis for span(S).

Example 2.

We performed Gauss –Jordan on columns and we learned the following: S is linearly dependent. S isn’t basis for span(S). But

is a basis for span(S).

Dimension span(S)=2 Example 3. Compression.

S is linearly independent because they are not multiple of each other. S is a basis for its span. Assume that the signal vectors are coming from span(S). For example a signal vector from this span

I need to send the signal to the other side.

I could send I could send what?

James says : Send instead . So we saved

66%. Let us test the system. We are on the other side and we received

. What was the original signal vector?

______Q: Find a different basis for span(S).

Q:Find a representation of in the new

basis. Big Question: Transition between two bases. Let

and

be two linear bases for the same vector space X. Let Then

and

Assume you know the first representation. How do you find the second representation? Or If you know a and b how can you find x and y(Really fast)? First notice that

Therefore it can be written in terms of the basis T.

Hence by __uniqueness____

We can write this in matrix notation:

transition matrix

In the case 2x2 transformation matrix