Transcription of Christian Parkinson UCLA Basic Exam Solutions: Linear ...
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Christian Parkinson UCLA Basic Exam Solutions: Linear ...
Christian ParkinsonUCLA Basic exam solutions : Linear Algebra1 Problem a Linear operator on a finite dimensional complex inner prod-uct spaceVsuch thatT T=TT . Show that there is an orthonormal basis ofVconsistingof eigenvectors T=TT , we prove this by induction on the dimension of the space thatToperates on. IfTisoperating on a 1-dimensional space, the claim is the claim holds for any normalToperating on ann 1 dimensional space(n 2). By the fundamental theorem of algebra, the characteristic polynomial ofT has aroot which is an eigenvalue ofT.
Christian Parkinson UCLA Basic Exam Solutions: Linear Algebra 2 Hence T(y j) 2ker(S) for each j.Further if a 1;:::;a k2C are such that a 1T(y 1) + + a kT(y k) = 0; Then T(a 1y 1 + + a ky k) = 0 so a 1y 1 + + a ky k2ker(T) so there are b 1;:::;b ‘2C such that a 1y 1 + + a ky k= b 1v 1 + b ‘v ‘ =) a 1y 1 + + a ky k b 1v 1 b ‘v ‘= 0: But these vectors form a basis for ker(S T) so in ...
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