10. Consider the basis for P, »{1,x,x+,x"}. Let T be a transformation T:P, , where T(x*)=...
(2) Let Pn [x] = {p € P[x] : degp <n}, where P[x] is the set of all polynomials. Let the polynomials li() defined by II;tilt - a;) i=0,1,...11 bi(T) = 11: a; - aj) where aj, j = 0,1,..., are distinct real numbers and aia . Show that (d) The change of basis transformation from the standard basis ', j = 0,1,...,n to l; () is given by the Vandermonde matrix (1 00 ... am 1 01 .01 1...
10. Let T : P P , be the linear transformation defined by T(P) = (a) What is the kernel of T? (b) According to the concept of the rank theorem, what is the dimension of the range of T? (C) (needs an idea from earlier in the semester) If we represent P, by coordinate vectors rela- tive to it's standard basis (1.1.1-.1') and P, by coordinate vectors relative to it's standard basis (1,1,1"), find the standard matrix A of...
Let T:P R^2 be defined by T(p(x)) = (p(1),p(-1)). (a) Find T(p(x)) where p(x) = 2 + 5x. (b) Show that T is a linear transformation. (C) Find the kernel of T. Explain why T is one-to-one. (d) Find the range of T. Explain why I' is onto. (e) Find T-1(3,7)
Consider the linear transformation T: "R" whose matrix A relative to the standard basis is given. A=[1:2] (a) Find the eigenvalues of A. (Enter your answers from smallest to largest.) (11, 12) = 2,3 |_) (b) Find a basis for each of the corresponding eigenspaces. B = X B2 = = {I (c) Find the matrix A' for T relative to the basis B', where B'is made up of the basis vectors found in part (b). A=
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 operator T : P) → given by where k 0 is a parameter (a) Find the matrix Tg,b representing T in the basis B (b) Verify whether T is one-to-one and whether or not it is onto. (c) Find the eigenvalues and the corresponding eigenspaces of the...
Let V P2(R) and let T V-V be a linear transformation defined by T(p)-q, where (x)(r p (r Let B = {x, 1 + x2, 2x-1} be a basis of V. Compute [TIB,B, and deduce if it is eigenvectors basis of
Question 4.1 (9 marks): Consider a basis B = {pl,p2.p3} of polynomials in P, , where pl :=1-x: p2 := x-x: p3 := 1+x: a Use the definition of coordinate vector to find the polynomial p4 in P, the vector of coordinates of which in the basis B is c4=(2,2,-2). b. Find the transition matrix StoB from the standard basis in P, to the basis B. What are the coordinates of the three standard coordinate vectors of the basis Sin...
Find the matrix [T], p of the linear transformation T: V - W with respect to the bases B and C of V and W, respectively. T:P, → P, defined by T(a + bx) = b - ax, B = {1 + x, 1 – x}, C = {1, x}, v = p(x) = 4 + 2x [T] C+B = Verify the theorem below for the vector v by computing T(v) directly and using the theorem. Let V and W...
Let p, (t) 6+t, P2(t) =t-3t, p3 (t) = 1 +t-2t. Complete parts (a) and (b) below. Use coordinate vectors to show that these polynomials form a basis for P2. What are the coordinate vectors corresponding to p, p2, and pa? P- Place these coordinate vectors into the columns fa matrix A. What can be said about the matrix A? O A. The matrix A forms a basis for R3 by the Invertible Matrix Theorem because all square matrices are...
1. Let L: P1(R) + P1(R) be a linear transformation given by L(a + bx) = a - b + (2a – b)x. Let S = {1, 2} and T = {1+x} be two basis for P1(R). (a) Find the matrix A of L with respect to basis S. (a) Find the matrix B of L with respect to basis T. (c) Find the matrix P obtained by expressing vectors in basis T in terms of vectors in basis (d)...