2. Let V = M₂ (R). Show that the symmetric bilinear form : VxV → R given by (X,Y)= tr(XY)is nondegenerate. (Hint. Suppose X = (ij) is such that Þ(X,Y)= 0 for all Y € V. Plug in Y = ek,the matrix whose (lk)-entry is 1, and all other entries are 0, to show that j = 0 for all i, j.)

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Answered: 2. Let V = M₂ (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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Consider the function ƒ (x₁, x₂ ) = 5 x² + 4x² defined on (x1, x2 ) = R². Which of the following is the Hessian matrix of f (x1, x2 )? off 5 [19] 8 -3 0 [] [39] 0 -4 [59] 5 x1 o[0] 4 x₂ 05 • [25]
T: V ---> V is a linear operator with a B and B' ordered bases for V a. Find the matrix representation for T relative to the ordered bases B and B' b. Find T(v), using a direct computation and using the matrix representation
7. Let T: R³ R³ be defined by (a) (b) (E) x1 + x2 + x3 = X1 X2 X3 [X1X2 X3] Find the matrix of linear transformation for T. T Find all vectors 7 € R³ such that T(x) = x.
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
Let f: R³ → R3 be a linear function so that -0-0-0-0-0-0- = 3 3 and f 1 3 3 . Find the matrix A such that f(v) = Av.
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).
Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|
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 -
Define T(f) = f′ from P4 to P4. (a) Prove T is linear. (b) Give the matrix for T. Is it invertible? Why or why not?
1. Which of the following transformations are linear? a. T: M→M defined by T (A) = A + I where A € M, and I is the nxn identity matrix. b. T: R² R³ defined by T(x, y) = (2x, x - y,0).
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)
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

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2. Let V = M₂ (R). Show that the symmetric bilinear form : VxV→R given by (X,Y)= tr(XY) (X,Y)=0 for all Y € V. Plug in Y = ek is nondegenerate. (Hint. Suppose X = (zij) is such that the matrix whose ((k)-entry is 1, and all other entries are 0, to show that zij = 0 for all i, j.)