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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

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

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CR: Review Exercises
Problem 41CR: Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the...
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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
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