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Question 2 over a field K. Let Mn(K) denotes the vector space of n × n matrices (a) Let U and W be the subspaces of V = M3(K) defined by U = {A € M3(K)|At = A}, W = {A € M3(K)|A² = −A} where At denotes the transpose of A. Prove that V = U W if K (b) Let A be given by = R and prove that VU W if K = F2. Compute the rank of A as A matrix over M₂(R) A matrix over M2(F3) A matrix over M₂(F2) 1 1 2 01 A = 0 -2 0 1 -1 R denotes the field of real numbers, Q denotes the field of rationals, and | Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures.

Question 2 over a field K. Let Mn(K) denotes the vector space of n × n matrices (a) Let U and W be the subspaces of V = M3(K) defined by U = {A € M3(K)|At = A}, W = {A € M3(K)|A² = −A} where At denotes the transpose of A. Prove that V = U W if K (b) Let A be given by = R and prove that VU W if K = F2. Compute the rank of A as A matrix over M₂(R) A matrix over M2(F3) A matrix over M₂(F2) 1 1 2 01 A = 0 -2 0 1 -1 R denotes the field of real numbers, Q denotes the field of rationals, and | Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 10E
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Question 2 over a field K. Let Mn(K) denotes the vector space of n × n matrices (a) Let U and W be the subspaces of V = M3(K) defined by U = {A € M3(K)|At = A}, W = {A € M3(K)|A² = −A} where At denotes the transpose of A. Prove that V = U W if K (b) Let A be given by = R and prove that VU W if K = F2. Compute the rank of A as A matrix over M₂(R) A matrix over M2(F3) A matrix over M₂(F2) 1 1 2 01 A = 0 -2 0 1 -1
Transcribed Image Text:Question 2 over a field K. Let Mn(K) denotes the vector space of n × n matrices (a) Let U and W be the subspaces of V = M3(K) defined by U = {A € M3(K)|At = A}, W = {A € M3(K)|A² = −A} where At denotes the transpose of A. Prove that V = U W if K (b) Let A be given by = R and prove that VU W if K = F2. Compute the rank of A as A matrix over M₂(R) A matrix over M2(F3) A matrix over M₂(F2) 1 1 2 01 A = 0 -2 0 1 -1
R denotes the field of real numbers, Q denotes the field of rationals, and | Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures.
Transcribed Image Text:R denotes the field of real numbers, Q denotes the field of rationals, and | Fp denotes the field of p elements given by integers modulo p. You may refer to general results from lectures.
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