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# Su11d - MAS 4105 Test 4(12 pts 1 a If u and v are vectors...

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MAS 4105 Test 4 (12 pts) 1. a. If u and v are vectors in an inner product space V , express the Cauchy-Schwartz Inequality in terms of u and v . b. If u and v are vectors in an inner product space V , express the Triangle Inequality in terms of u and v . c. If R 3 is considered as an inner product space with the standard inner product, give an ex- ample of an orthogonal set in R 3 which is not linearly independent. d. Let T be a linear operator on a vector space V ; give two conditions which will insure that the transformation is diagonalizable. e. Let T be a linear operator on a vector space V and let λ 1 and λ 2 be two distinct eigenvalues for T ; what is E λ 1 E λ 2 ? f. Let T be a linear operator on a vector space V and let λ be an eigenvalues for T ; if the multiplicity of λ is four, what are the possible dimensions of E λ 1 ?

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(8 pts) 2. Given the matrix A = 0 0 1 0 2 0 9 0 0 Find the eigenvalues and eigenvectors for the matrix. If the matrix A is diagonalizable, give the corresponding matrices Q and D .
(6 pts) 3. In the proof of the theorem below explain the underlined statement. Theorem 5.8. Let T be a linear operator on a vector space V , and let λ 1 , λ 2 , . . . , λ k be dis- tinct eigenvalues of T . For each i = 1 , 2 , . . . , k , let S i be a finite linearly independent subset of the eigenspace E λ i . Then S = S 1 S 2 ∪ · · · ∪ S k

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Su11d - MAS 4105 Test 4(12 pts 1 a If u and v are vectors...

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