for T:R R' relative to B. Given that [0), =%3D(a) Find the transition matrix P from B'to B.(b) Find P'and A'(the matrix for T relative to B).(c) Find [v),(d) Find [T(7)],(e) Find T (v) in two different ways.

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Answered: Let B = ((1,-3).(-2.5)) and B- ((1.–1).…

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 B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4, 4)} be bases for R², and let 23 = [33] 04 A = R2 relative to B. (a) Find the transition matrix P from B' to B. be the matrix for T: R² ->>> P = 6 9 [V] B [T(V)]B = (b) Use the matrices P and A to find [v] and [T(v)]B, where [V] B¹ = [-4 3]. -12 -1/3 -24 -96 4 -96 4 11 ← (c) Find P-1 and A' (the matrix for T relative to B'). 1/3
1. Let A = MB (f) be the matrix associated with the linear map f(x,y.z) = (3x-y+4z.y+z-x.x+y+z) relative to the bases B={u=(4,18,6), v= (17.1.0), w=(17,0,-5)} and B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0)) Then the first column of A is: OA-16 OB. 4 17 17 B 0 (3) 13 1 OC. 3 O D./ O E. 7-
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
Let V be the space spanned by f1= sin x and f 2= cos x. (a) Show that gj = 2sin x + cos x and g2 3cos x form a basis for V. (b) Find the transition matrix from B' = {g1, g2} to B= {f1, f2}. (c) Find the transition matrix from B to B'. (d) Compute the coordinate vector [h] g, where h= 2sin x -Scos and obtain [h]B'. X> | (e) Check your work by computing [h]g' directly.
Find a vector u E K such that, at the point (2,1, –1), the function f(, y, 2) = -y' + 222 + 2² has its greatest rate of increase in the direction of u. Ou= (-4, 2, 1) = (8,-3,-2) = (3,-2,-8) u (8, –3, 2) None of the others
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.
Let B = {ü,,ü,}, B'={v,v,} be two bases in R, where %| ü, %3D 3 (a) Find the transition matrix from B' to B. (b) Find the transition matrix from B to B'.
Let B={(1, 0), (0, 1)} and B'= {(1, 2), (2, 3)} be any two bases of R². Then verify PT],P= [T], where T(x, y)=(2x-3y, x +y) and P is the B'> transition matrix from B to B'.
2. Let a = 2 ---- X= -2 4 and y = 4 2 (a) Find P, the projection matrix along the span of a. (b) Find proja (x). (c) Find proja (y). 3. Let A be a nonsingular 2 x 2 matrix. (a) Find P, the projection matrix onto C(A). (b) Find C(A) and comment on the result of part (a).
5) PLEASE ANSWER EACH QUESTION, THANKS.
Find the transition matrix from the basis B = {(1,2,3) (1,0,1)(1,2,1)} to B' = {(1,1,0)(0,1,1)(1,1,1)}.

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Let B = ((1,-3).(-2.5)) and B- ((1.–1). (1.1)} be bases for T:R' R' relative to B. Given that [), - (a) Find the transition matrix P from B'to B. (b) Find P'and A'(the matrix for T relative to B). (c) Find [F],