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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online