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Let T: P2 + P, be a linear transformation for which T(1) = 3 - 2x,...
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:
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.
5. Let T: P2(R) R3 be a linear transformation such that T(1) = (-1,2, -3), T(1 + 3x) = (4,-5,6), and T(1 + x²) = (-7,8,-9). a. Show that {1,1 + 3x ,1 + x2} is a basis for P(R) (7pts) b. Compute T(-1+ 4x + 2x²). (3pts)
For each transformation below, find the closed form of the transformation. 1) Let T be a linear transformation from R$ to M22 (R) [i Let B=1 0:00 [. :] [11] [12] [0 ] Let C= 12 41 -17 -5 65 -27 92 Let M = be the matrix transformation of T from basis B to C 17 58 -15 -51 81 The closed form of the transformation is Tb 3-1 2) Let T be a linear transformation from P3(R) to...
Suppose T: M2,2-P2 is a linear transformation whose action on the standard basis for M2,2 is as follows: 1 0 0 1 0 0 0 0 T | = x2+x+2 = -x2+2x-3 x2–2x+4 T -2x2+x-4 0 0 o 0 1 Describe the action of T on a general matrix, using x as the variable for the polynomial and a, b, c, and d as constants. Use the '"' character to indicate an exponent, e.g. ax^2=bx+c. a b T = 0...
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
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}.
Let T:P1→P2T:P1→P2 be a linear transformation defined by T(a+bx)=3a−2bx+(a+b)x2.T(a+bx)=3a−2bx+(a+b)x2. (a) Find range(T)range(T) and give a basis for range(T)range(T). (b) Find ker(T)ker(T) and give a basis for ker(T)ker(T). (c) By justifying your answer determine whether TT is onto. (d) By justifying your answer determine whether TT is one-to-one. (e) Find [T(7+x)]B[T(7+x)]B, where B={−1,−2x,4x2}B={−1,−2x,4x2}.
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: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)]B, where B={−1,−2x,4x2} Please solve it in very detail, and make sure it is correct.