MATH 413/513, Section 001, Fall 2020
Quiz
MATH 413/513代考 For problems in this quiz you can use any of the theorems we covered in class.(2) All spaces here are fifinite-dimensional
General hints:
(1) For problems in this quiz you can use any of the theorems we covered in class.
(2) All spaces here are fifinite-dimensional
Please, submit no later than 1:00 p.m.
(1) (Ch. 7.A #12, 10 pt) Suppose T is a normal operator on V and 3 and 4 are eigenvalues of T. MATH 413/513代考
Prove that there exists vector ~v ∈ V such that ||~v|| = √ 2 and ||T~v|| = 5.
(2) (Ch. 7.A #17, 10 pt) Suppose T is a normal operator on V. Prove that null Tk = null T for any positive integer k. Hint: you are alllowed to use one of the spectral theorems.
(3) (Ch. 7.B #2, 15 pt) Suppose T is a self-adjoint operator on an inner product space, and that 3 and 2 are the only eigenvalues of T. MATH 413/513代考
Prove that there exists vector T2 − 5T + 6I = 0.
(4) (Ch. 7.B #6, 15 pt) For Math 413 only:
Prove that a normal operator on a complex inner product space is self-adjoint if and only if all its eigenvalues are real. MATH 413/513代考
(4) (Ch. 7.B #9, 15 pt) For Math 513 only:
Prove that any normal operator on a complex inner product space has a square root. Hint: the root does not have to be positive or uniquely defifined.
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