MAT 2401 Linear Algebra 4.4 Spanning Sets and Linear Independence .
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Transcript of MAT 2401 Linear Algebra 4.4 Spanning Sets and Linear Independence .
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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)
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
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.
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).
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)