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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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 Products
14) Let G = {(a;ij) E GL, (F)|a;j = 0 if i > j, and a11 = a22 = . = ann}, where F is a field, be the
of 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 upper
triangular matrices with 1's down the diagonal.
group](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0548f28d-2867-4ce8-91e2-809a65372be3%2F93378ad2-3e0f-4e8c-b6ca-05d96686f6c6%2Fc6ou8q_processed.png&w=3840&q=75)
Transcribed Image Text:5.4 Recognizing Direct Products
14) Let G = {(a;ij) E GL, (F)|a;j = 0 if i > j, and a11 = a22 = . = ann}, where F is a field, be the
of 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 upper
triangular matrices with 1's down the diagonal.
group
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