Let F be a field and U = Fx] be the vector space of polynomials in...
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Let F be a field and U = Fx] be the vector space of polynomials in one variable with coefficients in F. We define a linear operator D:U U by sending a polynomial to its derivative. More precisely, D is defined by the properties D(1) = 0, D(x) = 1, as well as the Leibniz rule, which makes use of the product of polynomials: D(fg) = (Df)g + fDg, Vf,g E F[x]. 1. Determine all eigenvalues and eigenvectors of D, and conclude that there is no nonzero element f e U satisfying the differential equation Df = f. Much in same way that we adjoin square roots of -1 to the real numbers in order to construct the complex numbers, we may adjoin a nontrivial solution to Df = f: we add an extra variable or "indeterminate" e, enlarging our vector space from U to V = F\x, e], the polynomials in two variables. We then extend our operator D to the larger vector space V by requiring the same conditions above (extending the Leibniz rule to all of Fx, e) as well as the new condition D(e) = e. We "abuse notation" by using the same name, D, for the original operator on U as well as its extension to V. 2. Determine all eigenvalues, eigenvectors, and generalized eigenvectors of the operator D on V. 3. Let Sc V be the subspace of solutions to the homogeneous differential equation f" – 4f" + 5f – 2f = 0, in other words, S = Null(D3 – 4D2 + 5D - 2). Find a basis for S putting D into Jordan form. Let F be a field and U = Fx] be the vector space of polynomials in one variable with coefficients in F. We define a linear operator D:U U by sending a polynomial to its derivative. More precisely, D is defined by the properties D(1) = 0, D(x) = 1, as well as the Leibniz rule, which makes use of the product of polynomials: D(fg) = (Df)g + fDg, Vf,g E F[x]. 1. Determine all eigenvalues and eigenvectors of D, and conclude that there is no nonzero element f e U satisfying the differential equation Df = f. Much in same way that we adjoin square roots of -1 to the real numbers in order to construct the complex numbers, we may adjoin a nontrivial solution to Df = f: we add an extra variable or "indeterminate" e, enlarging our vector space from U to V = F\x, e], the polynomials in two variables. We then extend our operator D to the larger vector space V by requiring the same conditions above (extending the Leibniz rule to all of Fx, e) as well as the new condition D(e) = e. We "abuse notation" by using the same name, D, for the original operator on U as well as its extension to V. 2. Determine all eigenvalues, eigenvectors, and generalized eigenvectors of the operator D on V. 3. Let Sc V be the subspace of solutions to the homogeneous differential equation f" – 4f" + 5f – 2f = 0, in other words, S = Null(D3 – 4D2 + 5D - 2). Find a basis for S putting D into Jordan form.
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Probability and Statistics
ISBN: 978-0321500465
4th edition
Authors: Morris H. DeGroot, Mark J. Schervish
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