Lecture19_Sec4-5_LinearIndependece

Lecture19_Sec4-5_LinearIndependece - -2 1 ² in M 22 are...

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MA 265 Lecture 19 Section 4.5 Linear Independence Recall The set W of all vectors of the form a b a + b is a subspace of R 3 . Example 1. Show that each of the following sets is a spanning set for W S 1 = 1 0 1 , 0 1 1 , S 2 = 1 0 1 , 0 1 1 , 3 2 5 1
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Definition of Linear Dependency The vectors v 1 , v 2 , · · · , v k in a vector space V are said to be linearly dependent if Remark v 1 , v 2 , · · · , v k are linearly independent if, v 1 , v 2 , · · · , v k are linearly dependent if, If S = { v 1 , v 2 , · · · , v k } , Example 2. Determine whether the vectors v 1 = 3 2 1 , v 2 = 1 2 0 , v 3 = - 1 2 - 1 , are linearly independent. MA 265 Lecture 19 page 2 of 4
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Example 3. Are the vectors v 1 = 2 1 0 1 , v 2 = 1 2 1 0 , v 3 = 0 - 3 - 2
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Unformatted text preview: -2 1 ² , in M 22 are linearly independent? Example 4. Determine whether the vectors v 1 = -1 1 , v 2 = -2 1 1 , are linearly independent. MA 265 Lecture 19 page 3 of 4 Theorem Let S = { v 1 , v 2 , ··· , v n } be a set of n vectors in R n . Example 5. Is S = { [1 2 3] , [0 1 2] , [3 0-1] } a linearly independent set of vectors in R 3 ? Theorem Let S 1 , S 2 be finite subsets of a vector space. Let S 1 be a subset of S 2 . Then • • MA 265 Lecture 19 page 4 of 4...
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