2. Let T be the linear transformation from P2 to R2 defined by 20 – 201...
Let T: P1 → P2 be a linear transformation defined by T(a + bx) = 3a – 2bx + (a + b)x². (a) Find range(T) and give a basis for range(T). (b) Find ker(T) and give a basis for ker(T). (c) By justifying your answer determine whether T is onto. (d) By justifying your answer determine whether T is one-to-one. (e) Find [T(7 + x)]], where B = {-1, -2x, 4x2}.
3. [20 marks] A linear transformation T: P2 + R’ is defined by [ 2a – b 1 T(a + bt + ct?) = a +b – 3c LC-a ] (1). [6 marks] Determine the kernel Ker T of the transformation T and express it in the form of a span of basis. Further, state the dimension of Ker T (2). [6 marks) Find the range Range T of the transformation T and express the range in the form of...
Problem #3: Let T: P2 P2 be the linear transformation defined by 7{p()) = (3x + 7) - that is 7(00+ cx + cox) = co + C (3x + 7) + C2(3x + 7)2 Find [7)with respect to the basis B = {1,x?). Enter the second row of the matrix 17 into the answer box below. i.e., if A = [718. then enter the values a1. 422, 223, (in that order), separated with commas. Problem #3:
let T: P2 --> R be the linear transformation defined by T(p(x))=p(2) a) What is the rank of T? b)what is the nullity of T? c)find a basis for Ker(T)
8. Let L: P2 → P be the linear transformation defined by Lar? +bt + c) = (a + b)t +(b - c). (a) Find a basis for ker L. (b) Find a basis for range L.
2. Let T: P2 + R2 be the linear transformation given by (a-6) T(a + bx + cx?) = | 16+c) Find ker T and im T.
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+ 4) = a +d+(6–c)n +(a–b+c+d)a? (1-1) (i) (3 marks) Find a basis for the T-cyclic subspace generated by (ii) (3 marks) Determine rank(T).
QUESTION 4 Let T R3-P2 be defined by T(a, b, c) - (a + b + e) +(a+b)a2 (4.1) Show that T is a linear transformation (4.2) Fınd the matrix representation [T]s, B, of T relative to the basıs in R3 and the basis in P2, ordered from left to right Determine the range R(T of T Is T onto? In other words, is it true that R(T)P2 Let x, y E R3 Show that x-y ker(T) f and only...
6. Let T P2 P be a linear transformation such that T P2P2 is still a linear trans formation such that T(1) 2r22 T(2-)=2 T(1) = 2r22 T(12 - )=2 T(x2x= 2r T(r2)2x (a) (6 points) Find the matrix for T in some basis B. Specify the basis that you use. (d) (4 points) Find a basis for the eigenspace E2. (b) (2 points) Find det(T) and tr(T') (e) (4 points) Find a basis = (f,9,h) for P2 such that...
Let T be the linear transformation from R3 into R2 defined by (1) For the standard ordered bases a and ß for R3 and IR2 respectively, find the associated matrix for T with respect to the bases α and β. (2) Let α = {x1 , X2, X3) and β = {yı, ys), where x1 = (1,0,-1), x2 = - (1,0). Find the associated (1,1,1), хз-(1,0,0), and y,-(0, 1), Уг matrices T]g and T12