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Unformatted text preview: MAS 4105 Test 4 (12 pts) 1. a. If u and v are vectors in an inner product space V , express the CauchySchwartz 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 1 and 2 be two distinct eigenvalues for T ; what is E 1 E 2 ? f. Let T be a linear operator on a vector space V and let be an eigenvalues for T ; if the multiplicity of is four, what are the possible dimensions of E 1 ? (8 pts) 2. Given the matrix A = 1 2 9 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 1 , 2 , . . . , 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 i . Then S = S 1...
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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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