5. Find the standard matrix for the linear operator T: R3 → R³ that first transform by theformula T(x, y, z) = (x – y + z, -x + y, y – z), then rotate the resulting vector anticlockwise about the z-axis through an angle 0 = 120°, and then project the resultingvector about zx-plane. Hence compute T(1,0,1).

Given that Tx,y,z=x-y+z, -x+y, y-z

Answered: - 5. Find the standard matrix for the…

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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4. Let W be the xy-plane in R³. (a) Find an orthonormal basis ū₁, 2 for W. (b) Construct the matrix Q = [ ₁ ü₂ ] and calculate the matrix QQT. (c) Consider the linear transformation T: R³ → R³ that projects each vector orthogonally onto the xy-plane. Note that T() = QQT. Use the matrix to calculate T(₁), T(2), and T(ē3). (d) Explain how your answers make sense geometrically.
Find the matrix A' for T relative to the basis B'. T: R² A' = X A' = 3 11/3 R², T(x, y) = (x - y, y - 2x), B' = {(1, -2), (0, 3)} -3 48 -16 -4 Find the matrix A' for T relative to the basis B'. ↓ 1 T: R² → R², T(x, y) = (-7x + y, 7x - y), B' = {(1, −1), (-1,5)} -72 24
Find the value of k for the linear operator T: R→R°, such ,3 that T(x,y,z) = (2x + ky,y, (k-2)y +z), be diagonalizable. Then, for the value of k found, determine the diagonal matrix similar to the matrix of the operator T on the canonical basis of R
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁0 T₂. T₁: R² R², T₁(x, y) = (x - 3y, 3x + 3y) T₂: R² → R², T₂(x, y) = (y, 0) A = A' = ↓ ↑ 00
Let f(u, v) = (u - v, 1,u + v) and g(x, y, z) = xyz. Find the entry a12(i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y) and D(g. f)(0, 1). (a) a12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(g)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) OD(g. f)(u, v) is a real-valued function of u and v. D(g. f)(u, v) is a 2 x 2 matrix. D(g. f)(u, v) is a 2 x 3 matrix. D(g f)(u, v) is a 3 x 2 matrix. 0 D(g f)(u, v) is a 1 x 2 matrix. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 2 x 1 matrix. D(f g)(x, y, z) is not defined. 0 D(f g)(x, y, z) is a 3 x 2 matrix. D(f g)(x, y, z) is a 2 x 2 matrix.
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁ 0 T₂. T₁: R² → R², T₁(x, y) = (x - 5y, 2x + 3y) T₂: R² R², T₂(x, y) = (0, x) A = Juma A' = SHE
5. Find the standard matrix A and A' for T = T2° T1 and T' = T1° T2, where T1:R2 → R3, T1(x,y) = (x,x+y,y) and T2:R3 → R², T2(x,y,z) = (0,y). Use standard basis vectors to derive your re- sults.
-3 Consider the matrix A = and | () the vector u = Let B = (A – A" )² and define vector V = Bu. If y = U2, write down the value of v .
Consider the function T : R → R², defined by T(r, y, 2) = (x – 2y + 2,1 – 2) (a) Write the matrix for T. (b) For the vectors u = (2,3, – 1), v = (-1,3,5) € R*, verify that T(3u + 4v) = 3T(u) + 4T(v). (c) For the unit vectors i = (1,0, 0), j = (0,1,0), k = (0,0, 1), write the matrix [T] = [T(i) T(j) T(k)]. (i.e. write the matrix [T] whose columns are the vectors T(i), T(j), T(k)) %3D
Let c₁ (t) = esti + 3 sin(t)j + t7k and c₂(t) = e−³¹i + 2 cos(t)j – 8t¹k. (Enter your solution as a single vector using the vector form (*,*,*). Use symbolic notation and fractions where needed.) -[c₁ (t) + c₂ (t)] = dt
Let B = {(1, 1, 0), (0, 1, 1), (1, 0, 1)} and B' = {(1, 0, 0), (0, 1, 0), (0, 0, 1)) be bases for R³, and let 1 2 1 2 A = P = 22555T 4 be the matrix for T: R3 R3 relative to B. [V] B -1 a) Find the transition matrix P from B' to B. 2 = 1 (b) Use the matrices P and A to find [v] and [T(v)]B, where [v] [-1 1 0]. [T(V)]B = 1
5. Find the standard matrix for the linear operator T: R3 → R³ that first transform by the formula T(x, y, z) = (2x – y + z, –x + 2y + z,x – z), then rotate the resulting vector anti clockwise about the y-axis through an angle 0 = 120°, and then project the resulting vector about yz-plane. Hence compute T(1, –1,1).

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