Problem 8 (Diagonalization). Let T : P₂ → P₂ be defined as, T(p(x)) = xp'(x). 1. Find the eigenvalues and eigenvectors of T. 2. Show that T is diagonalizable and write P2 as the sum of the eigenspaces of T. Problem 9 (Basis). Determine all the values of the scalar k for which the following four matrices form a basis for M2×2: A₁ = , A2 = k -3 0 , A3 = [ 1 0 -k 2 0 k " A₁ = . -1 -2 Problem 10 (Orthogonality). In this question, we will again see how orthogonality makes computations sim- pler. 1. Let u1,..., un be an (ONB) of a finite-dimensional inner product space V. Let v = c₁u₁ + ... + Сnun and w = d¹µ₁ + ... + dnUn be any two elements of V. Prove that (v, w) = c₁d₁ + ... + Cndn. 2. Write down the corresponding inner product formula for an orthogonal basis.

Problem 8 (Diagonalization). Let T : P₂ → P₂ be defined as, T(p(x)) = xp'(x). 1. Find the eigenvalues and eigenvectors of T. 2. Show that T is diagonalizable and write P2 as the sum of the eigenspaces of T. Problem 9 (Basis). Determine all the values of the scalar k for which the following four matrices form a basis for M2×2: A₁ = , A2 = k -3 0 , A3 = [ 1 0 -k 2 0 k " A₁ = . -1 -2 Problem 10 (Orthogonality). In this question, we will again see how orthogonality makes computations sim- pler. 1. Let u1,..., un be an (ONB) of a finite-dimensional inner product space V. Let v = c₁u₁ + ... + Сnun and w = d¹µ₁ + ... + dnUn be any two elements of V. Prove that (v, w) = c₁d₁ + ... + Cndn. 2. Write down the corresponding inner product formula for an orthogonal basis.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 77E

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