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).

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).

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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