Please provide answer in neat handwriting. Thank you
Please provide answer in neat handwriting. Thank you Let P2 be the vector space of all polynomials with degree at most 2, and B be the basis {1,T,T*). T(p(x))-p(kr); thus, Consider the linear operato...
Q3. Consider the vector space P, consisting of all polynomials of degree at most two together with the zero polynomial. Let S = {p.(t), p2(t)} be a set of polynomials in P, where: pi(t) = -4 +5, po(t) = -3° - 34+5 (a) Determine whether the set S = {P1(t).pz(t)} is linearly independent in Py? Provide a clear justification for your solution. (8 pts) (b) Determine whether the set S = {p(t),p2(t)} spans the vector space P ? Provide a...
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)...
Recall that P2 is the vector space of all polynomials of degree at most 2. Given U = Span({3+t?, t, 3t – 2,5t +t+1}), find the dimension of U as a subspace of P2.
3. P. is the vector space of all polynomials of degree n or less and the zero polynomial Define a derivative transformation T as follow: T. +P, T(+241 +0,2%) = 41 + 2121 (a) (10 Puan) Find the corresponding matrix for T. (b) (10 Puan) Choose your polynomial in P, and find the derivative of your polynomial by using the matrix in (a).
Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned by 8x−5x2+3, 2x-2x2+1 and 3x2-1. a) The dimensions of the subspace H is ___________? b) Is {8x-5x2+3, 2x-2x2+1, 3x2-1} a basis for P2? ________(be sure to explain and justify answer) c) A basis for the subspace H is {_________}? enter a polynomial or comma separated list of polynomials
15 5. Let P2 and Pz denote the vector space of polynomials of degrees atmost 2 and 3 respectively. Let T:P2 P3 be the transformation that maps a polynomial p(t) to the polynomial (t - 2)p(t). (a) Find the image of p(t) = t2, that is, find T(t2). (b) Show that T is a linear transformation. (c) Find the matrix of T relative to the bases B = {1,t, tº} and C = {1,t, t², tº}. (d) Is T onto?...
Q3. Consider the vector space B, consisting of all polynotninls of degree at most two together with the zero polynomial. Let S = {p(t).p2(t)} be a set of polynomials in P, where P.(t) = -2+3 pa(t) --21-24 + 3 (a) Determine whether the set S = {p(), pea(t)} is linearly independent in Py? Provide a clear justification for your solution. (8 pts) (b) Determine whether the set S = {p(t), pa(t)} spans the vector space B? Provide a clear justification...
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг + dr + ey + f | a,b,c, d, e, f E R} Let T be the linear operator on V defined by Find the Jordan canonical form of T 2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг...
Let V be the vector space of the polynomials of K [t] of degree less than 3, that is, the form p (x) = a2t2 + a1t1 + a0t0. Investigate the linear independence of the polynomials: p1 (t) = 1t2 + 0t1 + 1t0, p2 (t) = 2t2 + 2t1 + 0t0, p3 (t) = 4t2 + 1t1 + 3t0 for: *b) The modulo operation on 5 *c) The modulo operation on 7 a) K-R b) K c) K a)...