1. Let T be the set of all real 2x2 matrices with zero trace: T = {A € M2(IR) : tr(A) = 0} (a) Prove that T is a subspace of V = M2(R). (b) Construct a basis for T. Prove that your collection does in fact form a basis (Hint: You need to decide what a typical element of T looks like. What has to be true about the entries along the main diagonal of A for tr(A) = 0?) [1 -2 0 5 1 1 0 2. Let A = (a) Find a basis for the nullspace N(A) of A (indicate two sets in your answer: the set which forms your basis and the set which forms your nullspace). (b) Find a basis for the column space Col(A) of A (give two answers again as in (a): write down the basis for Col(A) and write down Col(A) itself).

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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1. Let T be the set of all real 2x2 matrices with zero trace: T = {A € M2(IR) : tr(A) = 0}
(a) Prove that T is a subspace of V = M2(R).
(b) Construct a basis for T. Prove that your collection does in fact form a basis (Hint:
You need to decide what a typical element of T looks like. What has to be true
about the entries along the main diagonal of A for tr(A) = 0?)
[1 -2 0 5
1 1 0
2. Let A =
(a) Find a basis for the nullspace N(A) of A (indicate two sets in your answer: the
set which forms your basis and the set which forms your nullspace).
(b) Find a basis for the column space Col(A) of A (give two answers again as in (a):
write down the basis for Col(A) and write down Col(A) itself).
Transcribed Image Text:1. Let T be the set of all real 2x2 matrices with zero trace: T = {A € M2(IR) : tr(A) = 0} (a) Prove that T is a subspace of V = M2(R). (b) Construct a basis for T. Prove that your collection does in fact form a basis (Hint: You need to decide what a typical element of T looks like. What has to be true about the entries along the main diagonal of A for tr(A) = 0?) [1 -2 0 5 1 1 0 2. Let A = (a) Find a basis for the nullspace N(A) of A (indicate two sets in your answer: the set which forms your basis and the set which forms your nullspace). (b) Find a basis for the column space Col(A) of A (give two answers again as in (a): write down the basis for Col(A) and write down Col(A) itself).
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