# a2 - CO350 Assignment 2 Wing Man Fok 20300650 Instructor...

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CO350 Assignment 2 Wing Man Fok 20300650 , May 21 2010 : - Instructor Mathieu Guay Paquet ( ) 1 a [ ] ,…, + , contrapositive Suppose the set a1 ak 1 is linearly dependent , ,…, + +… Then there exists scalars c1 ck 1 not all zero such that c1a1 + + + = …(*) ck 1ak 1 0 + ≠ . , ,…, , +… Note that ck 1 0 Otherwise since a1 ak is linearly independent c1a1 + + = = ∈ ,…, , . ckak 0 0 implies ci 0 for all i 1 n which is impossible *, Continue from + =- + - +…+ = = - + - ak 1 ck 1 1c1a1 ckak i 1k ck 1 1ciai + ,…, , + ∈ ,…, Since ak 1 is a linear combination of a1 ak ak 1 spana1 ak . This completes the proof ( ) 1 b ,…, . It suffices to show that a1 akcan be extended to a basis for Rm First check ≥ . ={ ,…, } = . > , + ∉ ∪ . , that m k Let A a1 ak and B ϕ When m k choose ak 1 spanA B Then + , ( ) . add the vector ak 1 to the set B by part a A B is linearly independent For , , finite dimensional spaces in this case Rm we can repeat this process until A B

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a2 - CO350 Assignment 2 Wing Man Fok 20300650 Instructor...

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