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MATH 413/513, Section 001, Fall 2020 Instructor: Leonid KunyanskyMATH 413/513代考 You are allowed to use the textbook and the notes of the lectures. A solution manual (or the”answer key”)for the textbook cannot be usedFinal test  MATH 413/513代考You are allowed to use the textbook and the notes of the lectures. A solution manual (or the ”answer key”) for the textbook cannot be used in any shape or form.(1) (20 pt) Suppose a, b, c, d are non-negative real numbers. Prove thata + 2b + 3c + 4d ≤ √30(a2  + b2  + c2  + d2).MATH 413/513代考(2) (25 pt) Consider vectors ˙v1 = (1, 1, 1, 1) and ˙v2 = (0, 1, 3, 4), and vector ˙u = (−2, 1, 3, 2).Prove  that  out  of  all  vectors  from  the  subspace  V span(˙v1, ˙v2), vector  w˙the closest vector to ˙u.MATH 413/513代考(3)(25 pt) Suppose dim V = 7, and T is an operator on V . At least four of theeigenvalues of T are distinct and not equal to zero. Consider now an arbitrary linear transformation mapping range T into null T (i.e. S ∈ L(range T, null T )). Prove that S cannot be injective.MATH 413/513代考(20 pt) The singular value decompostion for an operator T ∈ L(V ) has aform T ˙v = σ1 (˙v, ˙e1) f˙1  + σ2 (˙v, ˙e2) f˙2  +   + σ10 (˙v, ˙e10) f˙10, ∀˙v ∈ V,where ˙e1,  , ˙e10  and  f˙1,  , f˙10  are  orthonormal  bases  in  V.  Suppose  scalars  σ1, .., σ10  are  all nonzero.  Suppose also that span(˙e1, .., ˙e5) is invariant under T. Prove that the list consisting of six vectors f˙1, ˙e1, .., ˙e5 cannot be linearly independent.(5)(8+8+9pt) Suppose T is a normal operator on C3, MATH 413/513代考 with three eigenvalues equal to 1 + i,2 + i, and 3 + i, and eigenvectors ˙v1, ˙v2, and ˙v3.(a)What are the eigenvalues and eigenvectros ofT ∗?(b)What are the eigenvalues and eigenvectors ofT ∗T ?(c)What are the singular values of T?MATH 413/513代考(6)(5+10pt) Suppose T is a normal operator on a complex inner product  Suppose dim range T 1. Let us denote range T by U.(a)Prove that U is invariant under T (this is almosttrivial)(b)Prove that the restriction T |Uof T to subspace U is invertible.MATH 413/513代考(7)(5+10 pt) Extra-credit problem (no partial credit). Consider spaceR5.(a)Construct an operator T on R5 such that dim range T = 4, dim range T 2= 3, dim range T 3 = 2, dim range T 4 = 1, and T 5 0.(b)Using known properties of normal operators, prove that T is not a normal operator.MATH 413/513代考其他代写：program代写 cs作业代写 app代写  Programming代写 homework代写  考试助攻 finance代写 代写CS finance代写 java代写合作平台：essay代写 论文代写 写手招聘 英国留学生代写

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