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# problemset9 - MAS 213 Linear Algebra II Problem list for...

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MAS 213: Linear Algebra II. Problem list for Week #9. Tutorial on 9th November. This week’s topics: The eigenvalues and eigenvectors of an abstract linear operator. The multiplicity of an eigenvalue, and how it compares to the dimen- sion of the corresponding eigenspace. Diagonalizability for matrices (continued). Tutorial problems: Problem 1: (Problem 5.1.4, various parts, from [FIS].) In the parts below you are given a vector space V and a linear operator T : V V . Determine the eigenvalues of T , and find an ordered basis of V with respect to which the operator has a diagonal matrix representation. (i) V = P 1 ( R ) and T ( ax + b ) = ( - 6 a + 2 b ) x + ( - 6 a + b ). (ii) V = M 2 × 2 ( R ) and T ( A ) = A t + 2 · tr( A ) · I 2 . Problem 2: (Problem 5.2.7 from [FIS].) Let A = ( 1 4 2 3 ) M 2 × 2 ( R ) . Derive a formula for A n , where n N . Problem 3: Let A be a diagonalizable matrix. Show that every eigenvalue is either 1 or 0 if and only if A 2 = A . Hint: In this problem and the next one, you should use the theorem that a square matrix A

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problemset9 - MAS 213 Linear Algebra II Problem list for...

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