Abstract Algebra MATH 731
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This 1 page Class Notes was uploaded by Oswaldo Brown Sr. on Tuesday October 27, 2015. The Class Notes belongs to MATH 731 at University of Wisconsin - Milwaukee taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/230272/math-731-university-of-wisconsin-milwaukee in Mathematics (M) at University of Wisconsin - Milwaukee.
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Date Created: 10/27/15
MATH 731 FALL 2008 HOMEWORK SET 4 Due Friday7 October 3 at noon In doing this hornework7 you may make use of the results in the notes In particular7 you may use 1 The Spectral Theorern7 for rnatrices or for operators 2 The de nition of the adjoint of an operator and its basic properties Es lt v7wgt ltv wgt7 CW 545 45 A Let A E Prove that A is unitary if and only if A is normal and all eigenvalues of A have absolute value 1 B Suppose V is an inner product space and 15 V a V is a normal operator7 that is7 15 o 15 15 o 15 Hint the parts below are in the given order for a reason 1 Prove that H1511H for all 1 E V 2 Show that ker 15 ker 15 3 Show that if c is any scalar7 then 15 7 cidV is still a normal operator 4 Show that if 1 is an eigenvector for 15 with eigenvalue A then 1 is an eigenvector for 15 with eigenvalue A 5 Show that if 1110 are eigenvectors of 15 with different eigenvalues7 then 07w 0
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