Lect14 - Chapter 3 Vector Spaces Math1111 Linear...

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Unformatted text preview: Chapter 3. Vector Spaces Math1111 Linear Independence Motivation Recall that both { e 1 , e 2 , e 3 , ( 1 2 3 ) T } and { e 1 , e 2 , e 3 } are spanning sets for R 3 . The set { e 1 , e 2 , e 3 } is smaller in the sense that { e 1 , e 2 , e 3 } is a subset of { e 1 , e 2 , e 3 , ( 1 2 3 ) T } . Chapter 3. Vector Spaces Math1111 Linear Independence Motivation Recall that both { e 1 , e 2 , e 3 , ( 1 2 3 ) T } and { e 1 , e 2 , e 3 } are spanning sets for R 3 . The set { e 1 , e 2 , e 3 } is smaller in the sense that { e 1 , e 2 , e 3 } is a subset of { e 1 , e 2 , e 3 , ( 1 2 3 ) T } . Question Can we find a spanning set smaller than { e 1 , e 2 , e 3 } ? Chapter 3. Vector Spaces Math1111 Linear Independence Motivation Recall that both { e 1 , e 2 , e 3 , ( 1 2 3 ) T } and { e 1 , e 2 , e 3 } are spanning sets for R 3 . The set { e 1 , e 2 , e 3 } is smaller in the sense that { e 1 , e 2 , e 3 } is a subset of { e 1 , e 2 , e 3 , ( 1 2 3 ) T } . Question Can we find a spanning set smaller than { e 1 , e 2 , e 3 } ? Ans. NO . i.e. { e 1 , e 2 , e 3 } is a minimal spanning set . Chapter 3. Vector Spaces Math1111 Linear Independence Motivation Recall that both { e 1 , e 2 , e 3 , ( 1 2 3 ) T } and { e 1 , e 2 , e 3 } are spanning sets for R 3 . The set { e 1 , e 2 , e 3 } is smaller in the sense that { e 1 , e 2 , e 3 } is a subset of { e 1 , e 2 , e 3 , ( 1 2 3 ) T } . Question Can we find a spanning set smaller than { e 1 , e 2 , e 3 } ? Ans. NO . i.e. { e 1 , e 2 , e 3 } is a minimal spanning set . Question When is a spanning set minimal? Chapter 3. Vector Spaces Math1111 Linear Independence Motivation (Cont’d) Revisit the example: { e 1 , e 2 , e 3 , ( 1 2 3 ) T } is a spanning set for R 3 . { e 1 , e 2 , e 3 } is a small spanning set for R 3 . Chapter 3. Vector Spaces Math1111 Linear Independence Motivation (Cont’d) Revisit the example: { e 1 , e 2 , e 3 , ( 1 2 3 ) T } is a spanning set for R 3 . { e 1 , e 2 , e 3 } is a small spanning set for R 3 . ∵ ( 1 2 3 ) T ∈ R 3 , ∴ ( 1 2 3 ) T ∈ Span ( e 1 , e 2 , e 3 ) . Chapter 3. Vector Spaces Math1111 Linear Independence Motivation (Cont’d) Revisit the example: { e 1 , e 2 , e 3 , ( 1 2 3 ) T } is a spanning set for R 3 . { e 1 , e 2 , e 3 } is a small spanning set for R 3 . ∵ ( 1 2 3 ) T ∈ R 3 , ∴ ( 1 2 3 ) T ∈ Span ( e 1 , e 2 , e 3 ) . ∴ ( 1 2 3 ) T = α 1 e 1 + α 2 e 2 + α 3 e 3 for some scalars α 1 , α 2 , α 3 . α 1 e 1 + α 2 e 2 + α 3 e 3 +(- 1 )( 1 2 3 ) T = Chapter 3. Vector Spaces Math1111 Linear Independence Motivation (Cont’d) Revisit the example: { e 1 , e 2 , e 3 , ( 1 2 3 ) T } is a spanning set for R 3 . { e 1 , e 2 , e 3 } is a small spanning set for R 3 ....
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This note was uploaded on 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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Lect14 - Chapter 3 Vector Spaces Math1111 Linear...

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