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Unformatted text preview: MAS 4105 Test 4 1. (8 pts) Indicate whether the following are true or false. i. Let A ∈ M n × n ( R ); A is diagonalizable if for each eigenvalue of A , the dimension of the corresponding eigenspace is equal to the multiplicity of the eigenvalue. T F ii. Let A ∈ M n × n ( R ); A is diagonalizable if there exists n eigenvectors for A . T F iii. Let V be a finite dimensional inner product space of dimension n and let β be a set of k vectors in V with k < n ; the GramSchmidt process applied to β will yield a set of k distinct, orthogonal vectors. T F iv. Let V be a finitedimensional inner product space and let x,y ∈ V with both x and y nonzero vectors; then h x,y i > 0. T F v. Let V be a vector space, T a linear operator on V , and x an element of V ; if there exists a scalar λ such that T ( x ) = λx , then x is an eigenvector of T . T F vi. Let T be a linear operator on a finitedimensional vector space V , and let W be a Tinvariant subspace of V ; then the characteristic polynomial of T W divides the characteristic polynomial of the operator T . T F vii. Let T be a linear operator on a finitedimensional vector space V , and let W be the set of all elements x of V such that T ( x ) = x ; then W is a Tinvariant subspace of V . T F viii. Let V be a finitedimensional inner product space and let S be a set of orthogonal vectors in V ; then the vectors in S are linearly independent. T F 2. (10 pts) Let A = 2 4 4 4 4 6 . Find explicit representations for matrices Q,D, and Q 1 (with D diagonal) such that D = Q 1 AQ : 3. (10 pts) Prove that similar matrices have the same characteristic polynomial.3....
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 Spring '09
 RUDYAK
 Linear Algebra, Characteristic polynomial, 8 pts, Inner product space, 10 pts, Det

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