1. Let R₁ and R₂ be relations on a set A represented by the matrices./0 1 0MR,1 1 1 and MR₂ =/0 1 00 1 11 0 01 1 1/Find the matrices that representa) R₂0 R₂b) R₁ R₂c) R₁ R₂0d) Represent the above matrix, 8(c) with a digraph.

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Answered: 1. Let R₁ and R₂ be relations on a set…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.3: The Gram-schmidt Process And The Qr Factorization

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Let A and B be square matrices of order n satisfying, Ax=Bx for all x in all Rn. a Find the rank and nullity of AB. b Show that matrices A and B must be identical.
Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. Sec. 3.3,22b,32b Find the center Z(G) for each of the following groups G. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1. Find the centralizer for each element a in each of the following groups. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 Sec. 4.1,22 22. Find an isomorphism from the octic group D4 in Example 12 of this section to the group G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of Section 3.1. Sec. 4.6,14 14. Let G={ I2,R,R2,R3,H,D,V,T } be the multiplicative group of matrices in Exercise 36 of section 3.1, let G={ 1,1 } under multiplication, and define :GG by ([ abcd ])=adbc. Assume that is an epimorphism, and find the elements of K= ker . Write out the distinct elements of G/K. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of .

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1. Let R₁ and R₂ be relations on a set A represented by the matrices. 0 1 0 1 0 MR, 1 1 1 and MR₂ = 0 1 1 10 1 1 1/ Find the matrices that represent a) R₂0 R₂ b) R₁ R₂ c) R₁ R₂ d) Represent the above matrix, 8(c) with a digraph.