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Unformatted text preview: MAS 4105 Test 4 1. (8 pts) Let T be a linear operator on a finitedimensional 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 GramSchmidt 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 is an eigenvalue of T , then the multiplicity of is greater than or equal to the dimension of E . T F viii. If and are bases of V , then the characteristic polynomials of [ T ] and [ T ] are always equal. T F 2. (7 pts) Explain the underlined equality: Theorem 5.1. A linear operator T on a finitedimensional vector space V is diagonalizable if and only if there exists an ordered basis for V consisting of eigenvectors of T . Furthermore, if T is diagonalizable, = { v 1 , v 2 , . . . , v n } is an ordered basis of eigenvectors of T , and D = [ T ] , 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 finitedimensional vector space V . First suppose that T is diagonalizable. By definition, there exists an ordered basis = { v 1 , v 2 , . . . , v n } such that D = [ T ] is diagonal. For any v j , 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 = j v j where j = D jj . Hence by definition, v j is an eigenvector of T and j is the corresponding eigen value; therefore is an ordered basis consisting of eigenvectors of...
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This note was uploaded on 12/27/2011 for the course MAS 4105 taught by Professor Rudyak during the Spring '09 term at University of Florida.
 Spring '09
 RUDYAK

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