Im trying answer the attached for abstract algebra. I think I need a map from SL_nR to GL_nR, but Im not sure. Can you please assist, Thanks in advance. 5.4 Recognizing Direct Products14) Let G = {(a;ij) E GL, (F)|a;j = 0 if i > j, and a11 = a22 = . = ann}, where F is a field, be theof upper triangular matrices all of whose diagonal entries are equal. Prove that G = D × U,where D is the group of all non-zero multiples of the identity matrix and U is the group of uppertriangular matrices with 1's down the diagonal.group

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Description: Im trying answer the attached for abstract algebra. I think I need a map from SL_nR to GL_nR, but Im not sure. Can you please assist, Thanks in advance. 5.4 Recognizing Direct Products14) Let G = {(a;ij) E GL, (F)|a;j = 0 if i > j, and a11 = a22 = .

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14) Let G {(aij) E GL„(F)|a;j = 0 if i > j, and a11 a22 ann}, where F is a field, be the group of upper triangular matrices all of whose diagonal entries are equal. Prove that G = D x U, where D is the group of all non-zero multiples of the identity matrix and U is the triangular matrices with 1's down the diagonal. group of upper

14) Let G {(aij) E GL„(F)|a;j = 0 if i > j, and a11 a22 ann}, where F is a field, be the group of upper triangular matrices all of whose diagonal entries are equal. Prove that G = D x U, where D is the group of all non-zero multiples of the identity matrix and U is the triangular matrices with 1's down the diagonal. group of upper

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 13EQ

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Find the centralizer for each element a in each of the following groups. The quaternion group G={ 1,i,j,k,1,i,j,k } in Exercise 34 of section 3.1 (Sec. 3.1, #34). G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 (Sec. 3.1, #36). G={ I3,P1,P2,P3,P4,P5 } in Exercise 35 of section 3.1 (Sec. 3.1, #35). Sec. 3.1,34 34. Let G be the set of eight elements G={ 1,i,j,k,1,i,j,k } with identity element 1 and noncommutative multiplication given by (1)2=1, i2=j2=k2=1, ij=ji=k jk=kj=i, ki=ik=j, x=(1)x=x(1) for all x in G (The circular order of multiplication is indicated by the diagram in Figure 3.8.) Given that G is a group of order 8, write out the multiplication table for G. This group is known as the quaternion group. (Sec. 3.3,22a,32a, Sec. 3.4,2, Sec. 3.5,11, Sec. 4.2,8, Sec. 4.4,23, Sec. 4.5,40a, Sec. 4.6,3,11,16) Sec. 3.1,36 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, Sec. 4.1,22, Sec. 4.6,14) Sec. 3.1,35 35. A permutation matrix is a matrix that can be obtained from an identity matrix In by interchanging the rows one or more times (that is, by permuting the rows). For n=3 the permutation matrices are I3 and the five matrices. (Sec. 3.3,22c,32c, Sec. 3.4,5, Sec. 4.2,6) P1=[ 100001010 ] P2=[ 010100001 ] P3=[ 010001100 ] P4=[ 001010100 ] P5=[ 001100010 ] Given that G={ I3,P1,P2,P3,P4,P5 } is a group of order 6 with respect to matrix multiplication, write out a multiplication table for G.
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..
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