Let P be the set of real polynomials. Prove P is a vector space.
Let P2 be the real vector space of polynomials in a of degree at most 2, and let T be the real vector space of upper triangular 2 x 2 matrica b,cERThe vector space P2 is equipped with the inner product 〈p(x), q(x)-1 p(z)q(z) dr, and the vector space T is equipped with the inner product 〈A.B)=tr(AB), where tr denotes trace. Let L: P2→T be 1.p(z)dr]. Find L 0 c given by L(p(z)):-17(1) .CE :J ) 1 2 0 p(-1)...
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.
6. Let Pm (F) be the vector space of polynomials p(x) = ao + a1x + ... Amx" with coefficients in F and degree at most m, and let U be the set of even polynomials in P5(F): U := {p(x) € P5(F) | P(x) = p(-x)}. (a) Show that the list of vectors 1, x, x², x3, x4 + x, x + x spans P5(F). (b) Show that U is a vector subspace of P5(F) (c) Prove that there...
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...
Q3. Recall that P, is the vector space of all real polynomials of degree at most n. Determine whether the following subset of P, is a subspace: H = {p(t) € Pn such that p' (t)=0}, where p' (t) is the derivative of the polynomial p(t).
Q3. Recall that P, is the vector space of all real polynomials of degree at most n. Determine whether the following subset of P, is a subspace: H = {p(t) € Pn such that p'(t)=0}, where p' (t) is the derivative of the polynomial p(t).
Let P3 be the vector space of all real polynomials of degree at most 3. Determine whether S is a subspace of P3, where S
Question 2: For this question, consider the non-standard pairing on the space of real polynomials P given by g) = Lif(t)g(x).rº dr. (a) Prove that (,) defines an inner product on P. (b) Let O be the set of odd polynomials, i.e. f(r) € P such that f(x)= -f(-r). Show that is a subspace of P. (c) Explain why g() = 5x2 - 3 is in 0+ (the orthogonal complement of O with respect to (>). (d) Let P<2 denote...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
let P3 denote the vector space of polynomials of degree 3 or less, with an inner product defined by 14. Let Ps denote the vector space of polynomials of degree 3 or less, with an inner product defined by (p, q) Ji p(x)q(x) dr. Find an orthogo- nal basis for Ps that contains the vector 1+r. Find the norm (length) of each of your basis elements 14. Let Ps denote the vector space of polynomials of degree 3 or less,...