MAT 2401 Linear Algebra 4.4 Spanning Sets and Linear Independence

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Presentation transcript:

MAT 2401 Linear Algebra 4.4 Spanning Sets and Linear Independence

11/7 Friday Last day to withdraw from a class. Talk to me (or your advisor) if you have any questions. I and your advisor are here to help you make a informed decision.

Office Hours Changes

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 R 2 “smaller” than R 3 ? Why? Is R 2 “smaller” than P 2 ? 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

Example 0 (a)

Example 0 (b)

Example 0 (c)

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 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 0

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).