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.
ChapterP: Prerequisites
Section: Chapter Questions
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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
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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