2. Let V = Mn(R). Show that the symmetric bilinear form : V xV →R given by (X,Y) = tr(XY)is nondegenerate. (Hint. Suppose X = (xij) is such that (X,Y) = 0 for all Y E V. Plug in Y = eek;the matrix whose (lk)-entry is 1, and all other entries are 0, to show thatXij0 for all i, j.)%3D

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Answered: 2. Let V = Mn(R). Show that 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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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, z) and D(gof)(0, 1). (a) a 12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(gof)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) D(gof)(u, v) is a 1 x 2 matrix. D(gof)(u, v) is a 2 x 2 matrix. D(gof)(u, v) is a 3 x 2 matrix. D(gof)(u, v) is a 2 x 3 matrix. D(gof)(u, v) is a real-valued function of u and v. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is not defined. D(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 3 x 2 matrix. D(fog)(x, y, z) is a 2 x 2 matrix. D(fog)(x, y, z) is a 2 x 1 matrix.
For each of the following quadratic forms, write down the symmetric coefficient matrix A such that Q = vAv¹ and V = [X₁ X₂ Xn]. (a) Q (x, y, z) = -x² + 5 xy + 2y² + 2xz+3yz-3 z², (b) Q (x, y) = 2 x² + 4 xy + 4y², (c) Q (x, y, z) = -5x² + 5xy-3y² + 5xz+yz +4z², (d) Q (w, x, y, z) = −w² + 2 x² + 2 wy+xy − 5 y² − 5 w z - 4 xz - 5 y z + 3 z², (e) Q (x, y) = -2x² + 5xy-4y², A = A || A = A = = || II
8. Let L: R? → R² be the linear operator given by the rule L(r, y) (x + y, x – y) . and let B be the following ordered basis for R?: B = {(1,–2), (1, 1)} . Then the matrix [L]% of L with respect to B is which one of the following? [: 1 (a) [L]; = | (b) [피]B = | (c) [LE = | 1 -2 -2 (d) [L]; (e) [피뮴 = | -1 I3113
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 Then the first B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0) } column of A is: O A. 2 "B 1 3 B. - - 16 On () 13 1 C. D. E. 7 -5 3 4 17 17 3 -
5. Given that G(x, y) = (x² + 1, y²) and F(u, v) = (u + v, v²), compute the Jacobian derivative matrix of F(G(x, y)) at the point (x, y) = (1, 1).
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (2x − 2y+z, x − 4y+z, x-y-z) relative to the bases B={u=(5,16,1); v= (2, 1, –4); w=(7,17,2)} and B'={u'=(2,0,1); v' = (1,0,1); w'=(3,2,0)) Then the second column of A is: *) 0 O A. (0 O B. "B) °) SOC. O D. (6)
7). Given functions F(x, y) = (x²y x³ – y + 4, ) and G(u, v) = (u² + v, v) compute the Jacobian derivative matrix of function G o F at the point (1,0).
18.- Define the symmetric matrix corresponding to the following quadratic form: F(x, y, z)= x² + z²-2xy+xz - 2yz
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 O B. 13 1 4 17 17 °C) O D. (2 213 O E. "G) ņ m
7. Find the standard matrix for the linear transformation T. T:R²→R², T maps e¡ to e1 +4e2 but leaves e2 unchanged.

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2. Let V = Mn(R). Show that the symmetric bilinear form : V xV →R given by (X,Y) = tr(XY) is nondegenerate. (Hint. Suppose X: the matrix whose (lk)-entry is 1, and all other entries are 0, to show that xij = (xij) is such that (X,Y) = 0 for all Y E V. Plug in Y = eek, O for all i, j.)