(1) For the following linear transformations T find its matrix representation withrespect to the standard basis.(a) T(x, y, z) = (2y+z, x - 4y, 3x); find T(2, -3,1), T(-3,5,8),(b) T(x, y) = (3x + 2y, 4x - y); find T(2, 1), T(5,8),

Introduction: A function from one vector space to another that respects the underlying (linear)…

Answered: (1) For 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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(1) For the following linear transformations T find its matrix representation with respect to the standard basis.
Find the matrix of the linear transformation T from R2 - R² where T([1, 0]) = [1, –2] and T([2, 1]) = [2, 3] %3D (Note: matrix A = [T([1, 0]), T([0, 1])] 1 -2 (a) 1 (b) 1 (c) -2 (d) -2 3 7 3 Ob a C
Let B= (b,, b,, b,} and D= (d,, d,} be bases for vector spaces V and W, respectively. Let T :V- W be a linear transformation with the following properties. T(b,) = - 3d, +2d2, T(b2) = - 4d, + 4d2, T(b3) = 5d2 Find the matrix for T relative to B and D. The matrix for T relative to B and D isO.
Let v1, V2 and v3 be three linearly independent vectors in R3. (a) Find the rank of the matrix A = [(v1 – v2) (v2 – v3) (V3 – V1)]. (b) Find the rank of the matrix B = [(v1 + v2) (V2 + V3) (V3 + V1)].
Assume that T is a linear transformation. Find the standard matrix of T. T: R³ →R², T (e₁) = (1,6), and T (e₂) = (-4,3), and T (13) = (6, − 2), where e₁, 2, and e3 are the columns of the 3×3 identity matrix. A = (Type an integer or decimal for each matrix element.)
Consider the following linear transformation and basis. Find the standard matrix A for the linear transformation. A = Find the transition matrix P from B' to the standard basis B and then find its inverse. P = T: R² R², T(x, y) = (x - 5y, yx), B' = {(1, -2), (0, 3)) p-1 = A' = ↓1 ↓ 1 Find the matrix A' for T relative to the basis B'.
Let B= ((1, 3), (-2,-2)) and 8' = ((-12, 0), (-4, 4)) be bases for R2, and let be the matrix for T: R2 R2 relative to B. (a) Find the transition matrix P from B' to B. p= (b) Use the matrices P and A to find [v] and [7(v)le, where [vla [-1 31. [v] = [T(v)18- 11 p-1 11 (c) Find pand A' (the matrix for 7 relative to 8). 41 41 (d) Find [7(vil bwo ways. (7)=P(7) [vila Al- 11 11
3x₁ - x₂ [-x₁ + x₂] 1. (i) Show that 7 ([x₂]) = [³ T (ii) Find the matrix representation is a linear transformation. of T relative to the standard basis.
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…
SOLVE STEP BY STEP IN DIGITAL FORMAT PLEASE Through the point P(-2,-7) tangents are drawn to the ellipse 2x² + y² + 2x-3y - 2 = 0; determine the equations of the tangents and the points of tangency +

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(1) For the following linear transformations T find its matrix representation with respect to the standard basis. (a) T(x, y, z) = (2y + z, x − 4y, 3x); find T(2, −3, 1), T(−3, 5, 8), - (b) T(x, y) = (3x + 2y. 4x - y): find T(2.1). T(5.8).