MAT 2401 Linear Algebra 4 4 Spanning Sets

MAT 2401 Linear Algebra 4. 4 Spanning Sets and Linear Independence http: //myhome. spu. edu/lauw

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HW l l Web. Assign 4. 4 Part I & Part II Written Homework

Preview l Continue to examine the structure of vector spaces.

Questions l l l What is the “size” of a vector space? Is R 2 “smaller” than R 3 ? Why? Is R 2 “smaller” than P 2 ? Why?

Answers l l l 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 l l l 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 1 Determine whether u=(3, 1, 0) can be written as a linear combination of v 1=(1, 1, 2), v 2=(1, 0, -1), and v 3=(-5, -2, -1).

Q&A Q: What happens to the GJ elimination if u is not a linear combination of v 1, v 2, and v 3. A:

Spanning Set

Example 2 Let S={(1, 0, 0), (0, 1, 0), (0, 0, 1)} Determine whether S spans R 3.

Example 3 Let S={1, x, x 2} Determine whether S spans P 2.

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

The Span of a Subset

Example 5 Let S={(1, 0, 0), (0, 1, 0)} R 3. Interpret the geometric meaning of span(S).

Linear Independence

Linear Independence No one vector is a linear combination of the other vectors.

Example 6 Let S={(1, 0, 0), (0, 0, 2), (1, 0, 1)} Determine whether the set S is linearly independent or linearly dependent.

Example 7 Let S={(1, 1, 4), (1, 0, 3), (1, -1, 0)} Determine whether the set S is linearly independent or linearly dependent.

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