MAT 2401Linear Algebra

4.4 Spanning Sets and Linear Independence

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HW

WebAssign 4.4 Part I No Written Homework

Preview

Continue to examine the structure of vector spaces.

Questions

What is the “size” of a vector space?

Is R2 “smaller” than R3 ? Why? Is R2 “smaller” than P2 ? Why?

Answers

To answer these questions, we need to look into a few things…

Linear Combination (4.4) Spanning Set (4.4) Linear Independence (4.4) Basis (4.5) Dimension (4.5)

Answers

To answer these questions, we need to look into a few things…

Linear Combination (4.4) Not new Spanning Set (4.4) Linear Independence (4.4) ???? Basis (4.5) Dimension (4.5)

Linear Combination

Example 0

Representation of elements in

Example 0 (a)

Representation of elements in Let and Then every element can

be represented by a linear combination of and

Example 0 (a)

Representation of elements in Let and Then every element can

be represented by a linear combination of and

Write as a linear combination of and

Example 0 (b)

Let and Can every element be represented by a linear combination of and ?

See if you can write as a linear combination of and ?

Example 0 (c)

Let and Can every element be represented by a linear combination of and ?

See if you can write as a linear combination of and ?

Example 0 (c)

What is it so “bad” about and ?

Example 0

We used trial-and-error in this example.

It will not work for more complicated vector spaces.

We will illustrate a systematic method in the examples below.

Linear Combination

Example 1

Determine whether u=(3,1,0) can be written as a linear combination ofv1=(1,1,2), v2=(1,0,-1), and v3=(-5,-2,-1). 

Q&A

Q: What happens to the GJ elimination if u is not a linear combination of v1, v2, and v3.

A:

Spanning Set

Spanning Set

Spanning Set

Example 0

It can be easily checked that is a spanning set of . is a spanning set of .On the other hand, is not a spanning set of because

cannot be represented by a linear comb. of these 2 elements. (an counter example)

Example 2

Let S={(1,0,0), (0,1,0), (0,0,1)}Determine whether S spans R3.

Example 3

Let S={1, x, x2}Determine whether S spans P2.

Example 4

Let S={(1,1,4), (1,0,3), (0,1,1)}Determine whether S spans R3.

The Span of a Subset

𝑉

. . .. ……𝑆𝑠𝑝𝑎𝑛(𝑆)

The Span of a Subset

Example 5

Let S={(1,0,0), (0,1,0)} R3. Interpret the geometric meaning of span(S).