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Question 2: For this question, consider the non-standard pairing on the space of real polynomials P...
advanced linear algebra thxxxxxxxx Consider the complex vector space P4(C) of polynomials of degree at most 4 with coeffi- cients in C, equipped with the inner product ⟨ , ⟩ defined by 5. Consider the complex vector space P4(C) of polynomials of degree at most 4 with coeffi- cients in C, equipped with the inner product (, ) defined by (f, g)fx)g(xJdx. (a) Find an orthogonal basis of the subspace Pi(C)span,x (b) Find the element of Pi (C) that is...
Consider the three-dimensional subspace of function space defined by the span of 1, r, and a2 the first three orthogonal polynomials on -1,1. Let f(x) 21, and consider the subset G-{g(z) | 〈f,g〉 0), the set of functions orthogonal to f using the L inner product on, (This can be thought of as the plane normal to f(x) in the three-dimensional function space.) Let h(z) 2-1. Find the function g(x) є G in the plane which is closest to h(x)....
4 2-5 Notice that these polynomials form an Let P3 have the inner product given by evaluation at -3, -1, 1, and 3. Let po(t) = 2, P (t) = 4t, and act) = orthogonal set with this inner product. Find the best approximation to p(t) = tº by polynomials in Span{Po-P1:9). The best approximation to p(t) = tº by polynomials in Span{Po.P7.93 is
Exercise 13. For each pair of polynomials p(x), q(x) E P define (p, q) р(«)q(2) dx. -1 inner product (i) Prove that (p, q) defines on P3 an orthogonal (ii) Show that 1, х are (iii) Find the angle between 1 and 1 + x. Exercise 13. For each pair of polynomials p(x), q(x) E P define (p, q) р(«)q(2) dx. -1 inner product (i) Prove that (p, q) defines on P3 an orthogonal (ii) Show that 1, х are...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
2. Let P3 stand for the vector space of all polynomials in x with real coefficients and of the degree at most 3. (a) (1 mark) Show that the set E = {p(x) € P3 : p(3)=0}, is a subspace of P3. (b) (2 marks) Show that the collection of polynomials {(x - 3), (x – 3), (x-3)3} is a basis of E.
Topology (c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup norm of C[o, 1]. (i) Show that 5 is closed under pointwise multiplication, that is,if f,0€万 then fg P and, moreover, llfglloo for all f,g E P (c) Let P denote the vector subspace of C1O, 1] consisting of polynomial functions on [0,1. Let P be the closure of P in the sup...
Let V = R3[x] be the vector space of all polynomials with real coefficients and degress not exceeding 3. Let V-R3r] be the vector space of all polynomials with real coefficients and degress not exceeding 3. For 0Sn 3, define the maps dn p(x)HP(x) do where we adopt the convention thatp(x). Also define f V -V to be the linear map dro (a) Show that for O S n 3, T, is in the dual space V (b) LetTOs Show...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...