math 304 Test2sample

math 304 Test2sample - MATH 304505 Sample problems for Test...

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Unformatted text preview: MATH 304505 Sample problems for Test 2 Spring 2011 Any problem may be altered or replaced by a different one! Problem 1 (15 pts.) Let M2,2 (R) denote the vector space of 2 2 matrices with real entries. Consider a linear operator L : M2,2 (R) M2,2 (R) given by L x y z w = x y z w 1 2 . 3 4 Find the matrix of the operator L with respect to the basis E1 = 1 0 , 0 0 E2 = 0 1 , 0 0 E3 = 0 0 , 1 0 E4 = 0 0 . 0 1 Problem 2 (20 pts.) Find a linear polynomial which is the best least squares fit to the following data: -2 -1 0 1 2 x f (x) -3 -2 1 2 5 Problem 3 (25 pts.) Let V be a subspace of R4 spanned by the vectors x1 = (1, 1, 1, 1) and x2 = (1, 0, 3, 0). (i) Find an orthonormal basis for V . (ii) Find an orthonormal basis for the orthogonal complement V . 1 2 0 Problem 4 (30 pts.) Let A = 1 1 1 . 0 2 1 (i) Find all eigenvalues of the matrix A. (ii) For each eigenvalue of A, find an associated eigenvector. (iii) Is the matrix A diagonalizable? Explain. (iv) Find all eigenvalues of the matrix A2 . Bonus Problem 5 (15 pts.) Let L : V W be a linear mapping of a finite-dimensional vector space V to a vector space W . Show that dim Range(L) + dim ker(L) = dim V. ...
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