If T: R² → R³ is a linear transformation such thatthen the standard matrix of TisA =T([1])T([^])= 13 and T=

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Answered: If T: R² → R³ is a 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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Find the standard matrix representation for each linear operator L : R^2 → R^2 described below:(a) L rotates each vector x by 45◦ in the clockwise direction. (b) L reflects each vector x about the x1 axis and then rotates it 90◦ in the counterclockwise direction. (c) L doubles the length of x and then rotates it 30◦ in the counterclockwise direction. (d) L reflects each vector x about the line x2 = x1 and projects it onto the x1 axis.
3. Let T: R² R² be the linear transformation which first reflects points through the line X₁ = x₂ and then reflects points through the x₁-axis. a) Find the standard A matrix of T. b) Find all vectors v € R² such that v € Nul(A).
Write down the matrix A that represents the linear transformation f : R² → (a) R² (i.e. so that f(x) = Ax) that rotates anti-clockwise about the origin by an angle 7/2.
Compute the standard matrix for the linear map T : R² → R² which (1) rotates i by 0 = T, (2) stretches the x-component by 3 and (3) reflects the result of (2) about the T - axis.
Let a and B be the standard bases for R" and R" respectively. For each of the following linear transformations T; : R² → R", compute the matrix [T;%- 2a b () (a) T1 : R² → R³, T За + 46 a () - (*:) (C) (C) (2а + 3b —. a +b (b) Т, : R3 — R?, T, b (c) T3 : R³ → R', T3 = (2a + b – 3c). (d) T4 : R" → R", T4
Show that the transformation T defined by T(x,, x2) = (2x, - 3x2, x, + 3, 6x,) is not linear. If T is a linear transformation, then T(0) = and T(cu + dv) = cT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d. (Type a column vector.)
Find the image of the point (x, y) under the matrix transformation that first reflects the point with respect to the x-axis, then rotates the resulting point about the origin through an angle of T/3, and then reflects the resulting point with respect to the line y = x.
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).
Find the standard matrix for the linear transformation T(1, 9) — (2а — Зу, т — у, у — 4г) -3 1 -1 1 1 -4 2 1 -4 -3 -1 1 -3 -1 2 1 1 -3 -1 2.
Let S be a linear transformation from R³ to R² with associated matrix Let T be a linear transformation from R² to R² with associated matrix Determine the matrix C of the composition To S. C= 2 4-843 3]. A = B = 2 21.
10. Let T be the linear transformation from R3 to R$ that rotates every vector around the y-axis by radians, counterclockwise as viewed from the point (0,5, 0), and afterwards multiplies its length by 4. (a) Find the matrix of transformation T. (b) Calculate T((1, 2, 3)).
The cross product of two vectors in R³ is defined by Let = A = a₁ a2 a3 b1 [a2b3-a3b2] b2 = a3b1a1b3 ხვ La1b2-a2b1 H 1 Find the matrix A of the linear transformation from R³ to R³ given by T(z) = x *.

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If T: R² → R³ is a linear transformation such that then the standard matrix of Tis A = T([1]) T([^]) = 13 and T =