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Problem 3. Let Mnxn (R) denote the vector space of all n x n matrices with real entries. Let S be the subset of all symmetric matrices, i.e., S = {X € Mnxn (R) : X¹ = X}, and let A be the set of all anti-symmetric matrices, i.e., A = {X € Mnxn (R) : X¹ = −X}. We proved in Homework 2 that S and A are subspaces of Mnxn (R), and that Mnxn (R) SOA. = (a) Find bases for S and A. Note: You need to prove that your examples are in fact bases. (b) Use this information to compute the dimensions of S, A, and Mnxn(R).

Problem 3. Let Mnxn (R) denote the vector space of all n x n matrices with real entries. Let S be the subset of all symmetric matrices, i.e., S = {X € Mnxn (R) : X¹ = X}, and let A be the set of all anti-symmetric matrices, i.e., A = {X € Mnxn (R) : X¹ = −X}. We proved in Homework 2 that S and A are subspaces of Mnxn (R), and that Mnxn (R) SOA. = (a) Find bases for S and A. Note: You need to prove that your examples are in fact bases. (b) Use this information to compute the dimensions of S, A, and Mnxn(R).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 3. Let Mnxn(R) denote the vector space of all n × n matrices with real entries. Let S be the subset of all symmetric matrices, i.e., S = {X € Mnxn (R) : X¹ = X}, and let A be the set of all anti-symmetric matrices, i.e., A = {X € Mnxn (R) : X¹ = -X}. We proved in Homework 2 that S and A are subspaces of Mnxn (R), and that Mnxn(R) SO A. = (a) Find bases for S and A. Note: You need to prove that your examples are in fact bases. (b) Use this information to compute the dimensions of S, A, and Mnxn(R).
Transcribed Image Text:Problem 3. Let Mnxn(R) denote the vector space of all n × n matrices with real entries. Let S be the subset of all symmetric matrices, i.e., S = {X € Mnxn (R) : X¹ = X}, and let A be the set of all anti-symmetric matrices, i.e., A = {X € Mnxn (R) : X¹ = -X}. We proved in Homework 2 that S and A are subspaces of Mnxn (R), and that Mnxn(R) SO A. = (a) Find bases for S and A. Note: You need to prove that your examples are in fact bases. (b) Use this information to compute the dimensions of S, A, and Mnxn(R).
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