Let F = {0, 1, ...,p-1} be the field of order p (where p is a prime, and we perform arithmetic modulo p). (a) Prove that if A E M22(F), then det(A) 0 if and only if the columns of A are linearly independent. (b) Define GL(2, F) = {A € M22(F) | det(A) # 0} . %3D and recall that GL(2, F) is a group with the operation of matrix multiplication. Use (a) to calculate |GL(2,F)|. (c) Define D = {A € GL(2, F)| A is a diagonal matrix} ; T = {A € GL(2, F) | A is an upper triangular matrix}. Prove that T is a subgroup of GL(2, F); prove that D is a subgroup of T; calculate |D|| and calculate |T|. (d) Using the groups D and T from (c), consider the following function p:T → D a b 0 d a 0 0 d Prove that y is a homomorphism and calculate ker() and state its order.
Let F = {0, 1, ...,p-1} be the field of order p (where p is a prime, and we perform arithmetic modulo p). (a) Prove that if A E M22(F), then det(A) 0 if and only if the columns of A are linearly independent. (b) Define GL(2, F) = {A € M22(F) | det(A) # 0} . %3D and recall that GL(2, F) is a group with the operation of matrix multiplication. Use (a) to calculate |GL(2,F)|. (c) Define D = {A € GL(2, F)| A is a diagonal matrix} ; T = {A € GL(2, F) | A is an upper triangular matrix}. Prove that T is a subgroup of GL(2, F); prove that D is a subgroup of T; calculate |D|| and calculate |T|. (d) Using the groups D and T from (c), consider the following function p:T → D a b 0 d a 0 0 d Prove that y is a homomorphism and calculate ker() and state its order.
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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Transcribed Image Text:6. Let F = {0, 1,...,p-1} be the field of order p (where p is a prime, and we perform arithmetic
modulo p).
(a) Prove that if A e M22(F), then det(A) # 0 if and only if the columns of A are linearly
independent.
(b) Define
GL(2, F) = {A € M22(F) | det(A) # 0} .
and recall that GL(2, F) is a group with the operation of matrix multiplication. Use (a)
to calculate |GL(2, F)|.
(c) Define
D = {A € GL(2, F)| A is a diagonal matrix};
T = {A € GL(2, F) | A is an upper triangular matrix} .
Prove that T is a subgroup of GL(2, F); prove that D is a subgroup of T; calculate |D|
and calculate |T|.
(d) Using the groups D and T from (c), consider the following function
9:T → D
a b
d
Prove that p is a homomorphism and calculate ker(4) and state its order.
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