- Problem 3 Recall The Gram Schmidt Process From Linear Algebra If Vn N Is A Linearly Independent Sequence In An Inne 1 (144.3 KiB) Viewed 27 times
Problem 3. Recall the Gram-Schmidt process from linear algebra: If {vn}n-, is a linearly independent sequence in an inner product space (V, (:-)), define a new sequence {un}nen by uo = Vo uj = V1 = -U10 : : (V1, u) ||0|||2 : (Vn, uo) 10 ||0|||2 un = Vn (vn, u) |u1|2 U1 vn, un-1) Un-1 Jun-2 : : : Then the new sequence is orthogonal (i.e. (u;, u;) = 0 for i + j) and Span(10, U1, ... un) = Span(vo, V1, ... Vn) for all n. We get an orthonormal sequence {en} by putting en = un un a) Let V = C([0, 1], R) and define an inner product on V by (u, v) = ["u(a)u(x) dx . 2 ? Assume that we perform the Gram-Schmidt process on the polynomials Vo(C) = 1,01(C) = x, v2(x) = x²,..., Vn (2) = x", ... and get an ortho- normal sequence eo(2), C1(2), [2(x), ..., en(x), ... Find eo(2) and e1(). b) Let h € V and assume that (h, en) = 0) for n = 0, 1, 2, ... Show that (h, p) = 0 for all polynomials p. c) Show that (h, h) = 0, and conclude that h = 0. d) Let f V and put an = (f, en). Assume that g(x) = Emo Anen(x) is eV Ano continuous (the sum here is with respect to the norm | | generated by (;-), hence lim n700 ||g(x) – Eno Onen(2)] = 0). Show that g=f.