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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) 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).

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) 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).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.1: Operations With Matrices
Problem 77E
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Please write clearly. It is better if you can write by hand. Computers are hard to read.
we have know s and a are subspaces of Mn × n and s+a， please show （a）and （b）
thanks
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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