217 The space is C [0,21] with the inner product (f, g) = S fog(t)dt. Find...
2. Consider the inner product space V = P2(R) with (5,9) = . - f(t)g(t) dt, and let T:V + V be the linear operator defined by T(F) = xf'(x) + 2f (x). (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]2 is diagonal. If such a basis exists, find one.
Consider the inner product space V = P2(R) with (5,9) = { $(0)g(t) dt, and let T:VV be the linear operator defined by T(f) = x f'(x) +2f (x) +1. (i) Compute T*(1 + x + x2). (ii) Determine whether or not there is an orthonormal basis of eigenvectors ß for which [T]k is diagonal. If such a basis exists, find one.
4. Let L2(-π, π)) be the Lebesgue space of square integrable functions f: [-π, π] → C with inner-product, (f,g) =| f(t)g(t)dt (a) Show thatkt k e is an orthonormal system 2rZ s an orthonormal system (b) Let M be the linear span of (1, et, e). Find the point in M closest to the function [4 marks] 2π f(t) = t. [6 marks] 4. Let L2(-π, π)) be the Lebesgue space of square integrable functions f: [-π, π] →...
{D -3t0, 0t1 L'& f()g(t)dt. Let h(t) = 10] In C-3, 1, consider the inner product (f, g) (a) Find the function of the form ct that best approximates h(t) with respect to the above inner product (b) Find the function of the form cocit that best approximates h(t) with respect to the above inner product
dan Multiplication by t. 8. Find the following Laplace transforms using the formula L[t"f(t)] = (-1)", (a) [t3e-36] (b) C[(t + 2)2e'] (c) C[t(3 sin 2t - 2 cos 2t)] (d) L[tsin t] (e) C[t cosh 3t) (1) [(t-1)(t - 2) sin 3t] (g) [t3 cost] 9. Applying L[t"f(t)] = (-1)", , calculate (a) Sºte-3t sin t dt (b) Scºt?e-t cost dt recimento e contato Llegarsim 225 (-1)" IEC d'Fs) dsh
let two vectors be a(t) = e^t i + (sin 2t) j + t^3 k and b(t) = (e^-t , cos 3t, - 2 t^3) in euclidean three space R^3. Find d/dt [a(t) * b(t)].
Elementary Laplace Transtorms Y(S) = {f} -L e-stf(t)dt fc = C-'{F(s)} F(s) = {f} f(t) =-'{F(s)) F(s) = {f} -CS 1. 1 1 12. uct) le S> 0 S> 0 . s S 2. eat 1 13. ucOf(t-c) e-csF(s) S> a S-a n! 3. t",n e Z 14. ectf(t) F( sc) S> 0 sh+1 4. tP, p>-1 (p+1) S> 0 SP+1 15. f(ct) F). c>0 16. SFt - 1)g(t)dt F(s)G(*) 5. sin at S> 0 16. cos at 17. 8(t...
5. Let V be quadratic polynomials on the interval [-1,1], with the inner product 〈f,g):= | f(t)g(t)dt, and D VVfHf be the differentiation operator. (a) Find the Hermitian transpose (adjoint) D, which is determined by its action on a basis, by calculating D'(1), D*(x), D'(x2), explicitly. Find the eigenvalues and corresponding eigenfunctions of D* (c) Find (D) 5. Let V be quadratic polynomials on the interval [-1,1], with the inner product 〈f,g):= | f(t)g(t)dt, and D VVfHf be the differentiation...
Use the inner product <f,g>=∫10f(x)g(x)dx in the vector space C0[0,1] to find <f,g>, ||f||, ||g||, and the angle θf,g between f(x) and g(x) for f(x)=5x2−9 and g(x)=−9x+2.
6. (15 pts) Consider an inner product on the vector space P2[-1, 1] of polynomials of degree 2 or less in the closed interval [-1, 1], defined as follows: (f, 9) = | f(t)g(t) dt, for all f, ge P2[-1, 1]. Apply the Gram-Schmidt process to the basis {3, t – 2,t2 + 1} to obtain an {x1, X2, X3} = %3D orthonormal basis.