Let L: R → R² be defined byL(æ) = L12=Show that L is not a linear transformation by finding vectors æ, and , y such that L(x + y) + L(æ) + L(y):æ = (x1, T2)y = (Y1, Y2) :Prove your answer by calculating (for your choice of æ, Y):L(æ + y) =|andL(æ) + L(y)Note: This problem has infinitely many correct answers.

We have to prove that L is not a linear transformation.

Answered: Let L : R → R² be defined by L(æ) = L…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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8. Is there a linear transformation T: R³ → R² such that T(1, 0, 3) = (1, 1) and T(−2, 0, −6) = (2, 1) ? Explain.
(a) Let u and v be (fixed, but unknown) vectors in R". Suppose that T: R" → R" is a linear transformation such that T(u) = 6u + v and T(v) = 4u - 2v. Compute (T. T)(v), where TT is the composition of T with itself. Express your answer as a linear combination of u and v. (ToT)(v) = 10 u + -1 V Incorrect answer. Incorrect answer. (b) Let v and w be (fixed, but unknown) vectors in R", which are not scalar multiples of each others. Suppose that T: R" → R" is a linear transformation such that T(4v+3w) = -2v-2w and T(v+1w) = 5v+2w. Compute T(v) and express it as a linear combination of v and w. T(v) = 4 v + 2 W Incorrect answer. Incorrect answer.
Compute the following linear combinations for u = (1, 2), v = (4, -1), and w = (-1, 3). (a) u + w (b) u + 3v (c) v + w (d) 2u + 3v - w (е) —Зи + 4v — 2w
8. Let A = | 1 -3 -5 -7 7 5 (a) Find the image of u = and define T: R³ R2 by T(x) = Ax. 2 1 under T. (b) Find a vector x whose image under T is b = -12 12 Explain why your work makes sense.
Suppose T : C^2 → C^3 is a linear map so that T(1, 1) = (1, 0, 2) and T(2, 3) = (1,−1, 4). What is T(8, 11)?
2. Let T : R² → R² be a linear transformation such that: 1 -2 (i) T maps vector u = into the vector a = [ 5 3 (ii) T maps vector v = [³] into the vector b = [ Use the fact that T is a linear transformation to find T(3u +2v). -4
Use the function to find the image of v and the preimage of w. T(V3, V2) = ( V2, V1 + V2, 2v1 - V2 2 v = (2, 2), w = (-6/2, 0, -18) (a) the image of v (0,4,2) (b) the preimage of w (If the vector has an infinite number of solutions, give your answer in terms of the parameter t.) (2,36)
Check which of the followings are linear transformation (a) T : C→ C, such that T ( x + ί y ) = x (b) T : ?3 → ?4, s. t., T (a, b, c) = (a, a + b, a + b + c, c )
(5) Let y-(1) d u-(:) 3 and u = What is the orthogonal projection of y to the line through u and the origin, and what is the distance from y to the line through u and the origin.
Let P(2,3,1), I be the line that passes through Q(1,0, -1) in the direction of (1,1,2) and P: 2x+2y-z=0. Is it true that a(P, 1)

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