Find the determinant of the linear transformation T : P2 → P2 given byT(f) = -3f – 2fwhere P, denotes the vector space of polynomials of degree up to 2.Hint: The determinant of T is the determinant of the matrix associated to T with respect to some basis of P2.det(T)

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Answered: Find the determinant of the 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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Assume that T is a linear transformation. Find the standard matrix of T. T: R2→R*, T(e,) = (8, 1, 8, 1), and T(e2) = (-6, 9, 0, 0), where e, = (1,0) and e, = (0,1).
Consider the linear transformation T: R3 R defined by T(x, y, z) = (x – 3y + 2z, 3x + y + 5z, -2x + 6y – 4z). Find the standard matrix of T.
The given T is a linear transformation from R² into R². Show that T is invertible and find a formula for T¹. T(x1,x2) = (4x₁8x2, - 4x₁ + 9x2) To show that T is invertible, calculate the determinant of the standard matrix for T. The determinant of the standard matrix is (Simplify your answer.)
Let T be the linear transformation defined by T(x, y) = (-9y,4x + y, 5y – 8x, 2y – 7x). Find its associated matrix A. A =
Cross product is a linear transformation from R³ →R³ You can define it as X1 X2 X3 X Yı Y2 Y3 e₁ X1 Y1 x2 Y2 Y3 = = det( e2 → ez X3 Note that this is the determinant of a matrix that has vectors as some of the entries. You can look at section 3.6 in the textbook for alternate definitions. Define the linear transformation Tas T(x) = x x 2 3 a. Compute T(ei), T(₂), Te) b. Use your answer from a. to write the standard matrix of the linear transformation T c. Use your answer from b. to compute T( 4 H 3
The given T is a linear transformation from R² into R². Show that T is invertible and find a formula for T¹. T(x1,x2) = (6x₁-9x2,- 6x₁ +5x₂) To show that T is invertible, calculate the determinant of the standard matrix for T. The determinant of the standard matrix is (Simplify your answer.)
Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…
defined by Let Let ƒ : R² → R² be the linear transformation [f] B = B C f(x) = = = 3 2 -2 - -3 x. be two different bases for R². Find the matrix [f] for f relative to the basis B in the domain and C in the codomain. {(-1,2), (3,-7)}, {{-1, -2), (3, 7)},

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Find the determinant of the linear transformation T: P2 + Pa given by T(f) = -3f – 2f' where P2 denotes the vector space of polynomials of degree up to 2. Hint: The determinant of T is the determinant of the matrix associated to T with respect to some basis of P. det(T)