Consider the following linear transformation of R³:T(F₁, T2, T3) =(−2 ⋅ x₁ − 2 ⋅ x2 + x3, 2 ⋅ x₁ + 2 ⋅ x2.(i) What is the matrix A of T with respect to the standard basis for R³?A =(ii) Which of the following is a basis for the kernel of T?O(No answer given)O{(1, 0, -2), (-1, 1, 0)}O {(2, 0, 4), (-1, 1, 0), (0, 1, 1)}O{(-1, 1, -3)}○ {(0, 0, 0)}−x3, 6₁ + 6-₂-3.23). (iii) Which of the following is a basis for the image of T?O(No answer given)O {(2,0, 4), (1, -1,0)}O {(1, 0, 2), (-1, 1, 0), (0, 1, 1)}O {(-1, 1,3)}{(1, 0, 0), (0, 1, 0), (0, 0, 1)}

Tx1, x2, x3=-2x1-2x2+x3, 2x1+2x2-x3, 6x1+6x2-3x3

Answered: Consider the following linear…

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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Let TR5 R³ be the linear transformation given by X1 x2 T x3 x4 X5 x1 + 2x2 x3 + 2x4x5 2x14x22x3 + 5x4 + x5 2x14x22x3 + 6x5 (a) Find the image of T and a basis for it. (b) Find the kernel of T and a basis for it. (c) Find the rank and nullity of T. Is the dimension of the domain of T equal to the sum of the rank and nullity that you found?
(1) For the following linear transformations T find its matrix representation with respect to the standard basis.
Suppose B = {b1, b2} is a basis for V. C = {C₁, C₂, C3 is a basis for W. Let T: V→ W be a linear transformation with the property that: T (b₁) = 3c₁ - 2₂ + 5c3 y T (b₂) = 4¯₁ + 7C₂-C3 Find the matrix M for T relative to B and C. Determine the kernel and image of T.
Let T : P2 → M2x2 be the linear transformation defined by T(ax? + bx + c) |0 a (a) Find the matrix for T with respect to the standard bases of P2 and M2x2- (b) Find the matrix for T with respect t B basis for M2x2. {1, x+1, x²+2x+1} and the standard
Find the standard matrix of the linear transformation T: R² → R³ given by T(x₁, x2) = (2x₁ - 3x2, x₁ x1 1 1 [23 -3 -2 0 1 0 0 2 1 0 -2 0 0 2 -3 1 -2 2 -3 H 1 -2 1 -3 Го 2x2, x₂).
Show that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.
Consider the following linear transformation T: R5 → R3 where T(x1, X2, X3, X4, Xs) = (X1-X3+X4, 2x1+X2-X3+2x4, -2x1+3x3-3x4+Xs) (a) Determine the standard matrix representation A of T(x). (b) Find a basis for the kernel of T(x). (c) Find a basis for the range of T(x). (d) Is T(x) one-to-one? Is T(x) onto? Explain. (e) Is T(x) invertible? Explain
[2] such that 7(6₁)=56, +46, and 7(6₂)=35, +56₂. Let == B= and ba = (a) The 28-matrix of T (in other words, the matrix of T relative to the basis 33) is 5 3 A= The set = {1,5} is a basis for R². Let T: R2 R² be a linear transformation → 5 (b) The standard matrix of 7 (in other words, the matrix of T relative to the standard basis for R²) is -1 You have answered these types of questions (in b) before. Can you now use coordinates and the A-SBS-¹ framework? Which technique do you prefer?
A Let T be the linear transformation of P2 (F) defined by the formula T(P(x)) = (x + 2)P'(x) − P(x) a) Find the matrix of T in the standard basis (1, x, x²). :
Consider the linear transformation T: C→C of the last problem. Find the matrix T that represents it in each of the following basis: a) {1, i}, b) {1+i, 1+2i}.
(a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 4 9 -1 X1 + 2x2 - x3 X1 X1 X1 T| x2 4x1 + 9x2 - x3 i.e. T x2 X2 2x1 + 7x2 + 7x3) X3 2 7 7 X3 (i) Find a basis for the kernel of T. (ii) Find a basis for the image of T. (iii) Does the image of T include every vector in R'? Justify your answer. (You are not required to verify that your answers to (a) (i), (a) (i) are bases.) (b) Consider the following subset of R: X1 S = x2 ER: x2 + 4x3 = 2 Does there exist a real linear map, from R to R' say, such that the set S is the image of the real linear map? Justify your answer.

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