201s05fin

# 201s05fin - B U Department of Mathematics Math 201 Matrix...

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B U Department of Mathematics Math 201 Matrix Theory Spring 2005 Final Exam This archive is a property of Bo˘ gazi¸ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-proﬁt service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-proﬁt purpose may result in severe civil and criminal penalties. 1.) Consider the set of 2 × 2, symmetric matrices. a) Show that this is a subspace of all 2 × 2 matrices. b) Write a basis for this space and ﬁnd its dimension. c) Consider the set of n × n symmetric matrices. What is the dimension of this space? (You should explain your answer.) Solution: a) Let S 2 × 2 denote the set of 2 × 2 symmetric matrices. Let ± a 1 b 1 b 1 d 1 ² and ± a 2 b 2 b 2 d 2 ² be any two elements of S 2 × 2 . Then, for c R , we have ± a 1 b 1 b 1 d 1 ² + c ± a 2 b 2 b 2 d 2 ² = ± a 1 + ca 2 b 1 + cb 2 b 1 + cb 2 d 1 + cd 2 ² S 2 × 2 . Hence S 2 × 2 is a subspace of all 2 × 2 matrices. b) Clearly, ³± 1 0 0 0 ² , ± 0 0 0 1 ² , ± 0 1 1 0 ²´ is a basis for S 2 × 2 as its a linearly independent set spanning S 2 × 2 . Thus, dimension of a vector space being equal to the number of elements in a basis, we get that dim S 2 × 2 = 3. c) It is clear that to ﬁnd the number of basis elements of an n × n symmetric matrix we need to count the number elements on the upper triangular part including the diagonal. (This is because for a symmetric matrix elements below the diagonal are determined by

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201s05fin - B U Department of Mathematics Math 201 Matrix...

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