linear algebra 2.pdf

# linear algebra 2.pdf - Preliminary Exam in Linear Algebra...

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Preliminary Exam in Linear Algebra April, 2006 Show all your work. 1. Let P k denote the set of all polynomials of degree k or less on the unit interval 0 x 1. This is a k + 1 dimensional linear space since any linear combination of functions in P k is again in P k . It has a basis consisting (for example) of the monomial functions 1 , x, x 2 , x 3 , . . . , x k . These functions span P k and are linearly independent, i.e., the only linear combination of these functions that yields the zero function is the trivial linear combination with all coefficients equal to zero. (a) Define the inner product of two functions f ( x ) and g ( x ) defined on the interval [0 , 1] by h f, g i = Z 1 0 f ( x ) g ( x ) dx. We say f and g are orthogonal if h f, g i = 0. Define the 2-norm of f by k f k 2 = p h f, f i . Find the least squares approximation to the function f ( x ) = x 3 by a linear polyno- mial p ∈ P 1 on the interval [0 , 1], i.e., find the p ( x ) that minimizes k p - f k 2 . (b) Consider the set of functions S = { p ∈ P 2 : R 1 0 p ( x ) dx = R 1 0 p 0 ( x ) dx } . Show that this is a linear subspace of P 2 , determine its dimension and find a basis for S . 2. Let A = 7 0 0 0 3 1 0 6 2 . (a) Determine the eigenvalues of A and a basis for each eigenspace.

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• Fall '16
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