MAS4105Su10d

# MAS4105Su10d - MAS 4105 Test 4 1(8 pts Let T be a linear...

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Unformatted text preview: MAS 4105 Test 4 1. (8 pts) Let T be a linear operator on a finite-dimensional vector space V . Indicate whether the following are true or false. i. If V is an inner product space, then every basis of V can be transformed into an orthonormal basis of V . T F ii. If f ( t ) is the characteristic polynomial of T , then f ( T ) = 0. T F iii. The Gram-Schmidt process generates an orthonormal set of vectors. T F iv. If v 1 and v 2 are eigenvectors of T , then v 1 + v 2 is also an eigenvector. T F v. If V is an inner product space, x, y, z ∈ V , and h x, y i = h x, z i for all x ∈ V , then y = z . T F vi. If V is an inner product space and S is an orthogonal subset of V consisting of nonzero vectors, then S is linearly independent. T F vii. If λ is an eigenvalue of T , then the multiplicity of λ is greater than or equal to the dimension of E λ . T F viii. If β and γ are bases of V , then the characteristic polynomials of [ T ] β and [ T ] γ are always equal. T F 2. (7 pts) Explain the underlined equality: Theorem 5.1. A linear operator T on a finite-dimensional vector space V is diagonalizable if and only if there exists an ordered basis β for V consisting of eigenvectors of T . Furthermore, if T is diagonalizable, β = { v 1 , v 2 , . . . , v n } is an ordered basis of eigenvectors of T , and D = [ T ] β , then D is a diagonal matrix and D jj is the eigenvalue corresponding to v j for 1 ≤ j ≤ n . Proof. Let T be a linear operator on a finite-dimensional vector space V . First suppose that T is diagonalizable. By definition, there exists an ordered basis β = { v 1 , v 2 , . . . , v n } such that D = [ T ] β is diagonal. For any v j ∈ β , the image of v j under T is given by T (v j ) = n X i=1 D ij v i = D jj v j = λ j v j where λ j = D jj . Hence by definition, v j is an eigenvector of T and λ j is the corresponding eigen- value; β therefore is an ordered basis consisting of eigenvectors of...
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MAS4105Su10d - MAS 4105 Test 4 1(8 pts Let T be a linear...

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