Exercise 4 5 points Let a b be real numbers Consider the following matrix A B B

Exercise 4 5 points let a b be real numbers consider

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Exercise 4 (5 points):Leta, bbe real numbers. Consider the following matrix:A=0BB@1ab001ab1CCAFor which valuesaandbisAinvertible? For these values, write downA-1in terms ofaandb(simplify all expressions). 0 0 1 a 0 0 0 1 Solution:
Exercise 5 (8 points):Which of the following are subspaces ofR2? If you think thegiven set is a subspace, prove it. If you think the given set is not a subspace, show thatit isn’t.(1) The setVof vectors [x1x2] such that|x1|=|x2|.(2) The setVof vectors [x1x2] such thatx1-2x2= 0.Which of the following maps are linear transformations? (Justify your answer) (3)T:R2!R2defined byT([x1x2]) = [x1·x2x1+x2].(4)T:R2!R2defined byT([x1x2]) =x1+x2x1-x2.Solution:
Exericse 6 (8 points):Consider the following matrix:A=2412342345345635(1) Find a basis for Ker(A). What is dim(Ker(A))?(2) Find a basis for Im(A). What is dim(Im(A))? Solution:
Exercise 7 (8 points):LetVP2(R) be the set of polynomialsf(X) =a0+a1X+a2X2such thatf(1) = 0.(1) Show thatVis a subspace ofP2(R).(2) Find a basis ofV. What is dim(V)? Solution:
Exercise 8 (8 points):Consider the vectorsv1=266410013775,v2=26641-1113775,v3=2664101-13775(1) Show thatv1, v2, v3are linearly independent inR4.(2) Construct an orthonormal basis ofV= Span(v1, v2, v3). Solution:

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