10w_math115a_midterm_solutions

10w_math115a_midterm_solutions - Math 115A Midterm...

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Unformatted text preview: Math 115A Midterm Solutions Prof. Yehuda Shalom TA: Darren Creutz Date: 12 Feb 2010 1. Follow the instructions at the end of the following assignment. (i). State (without proof) the Replacement Theorem. (ii). Use the Replacement Theorem to show that every basis of a finitely generated vector space V has the same number of elements. (iii). Show that if S = { v 1 . . . v n } is an ordered set of vectors which is linearly dependent then there exists some vector in S which is a linear combination of the previous ones in the sequence. (iv). Use (i) and (iii) to show that a subspace of a finitely generated vector space is itself finitely generated. Instructions: Besides using the basic definitions and properties of vector spaces, you are only allowed to use here the two results appearing in (i) and (iii) (the latter needs to be proved, but can be used in (iv) also if you cannot prove it). Do not quote without proof any other non-trivial theorem, and in particular, you may not use at all the notion of dimension (or any result pertaining to it) without proof (in fact, any use of the word dimension in your writing would indicate a serious problem in your understanding of what is required here). If you find part (iv) too challenging, do not waste much time on it and move on. Proof. (i) Replacement Theorem: Let V be a vector space generated by a finite set S and let L be a linearly independent subset of V . Then # L # S and there exists a subset H of S containing exactly # S- # L vectors such that L H gen- erates V . (ii) Let V be finitely generated. Let B 1 and B 2 be bases of V . By the Re- placement Theorem, B 1 extends to a basis B 1 by adding # B 2- # B 1 elements of B 2 to B 1 . But B 1 is already a basis so is linearly independent, and B 1 is also linearly independent so we must not have added any vectors. Then # B 2 = # B 1 ....
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10w_math115a_midterm_solutions - Math 115A Midterm...

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