(2) (a) For any O E [ 0 21] let -sino Cose x For Cosce sino 1² [ a b ] simplity any matrix A АХ 052 If A = and [33]... B =[2] C], find X-sored that A(x+B) = C. Q 2 (C) Let S be the set of matrices of the form As a a2 ag where arbitrary real numbers. Show there exists a unique matrix E in s such that A EA for all o in وگرنه...
Below is the transformation matrix between cylindrical and rectangular coordinates: P cos sino 0 i 0 = -sino cosy 09 2 0 0 1 When we found the velocity and acceleration in cylindrical coordinates, we had to find how each of the unit vectors changed in time. do do For this problem, just find de and 4 di
Let X and Y be defined by: X = cos Y = sino Where is a uniformly distributed r.v. between 0 and 21. a) Show that X and Y are uncorrelated. b) Show that X and Y are not independent.
Find sin , cos , and tan - O A. sino= _ . cosO=- , tan 0= - 13 O B. sino= - ], COSO = 3, tan o=1/3 o c. sino=- , coso - ., tan og OD. sino - 13. COSO = 1, tan 0=13 Find the exact value of sin 510º. O B. v O A. OC. O D. Find the exact value of tan 111. OA OB. Z OD. 13 OC. 2 Suppose that there is...
2. (a) Find a 2 x 2 matrix A such that AP + 12 = 0. (b) Show that there is no 5 x 5 matrix B such that B2 + 15 = 0. (c) Let C be any n xn matrix such that C2 + In 0. Let l be any eigenvalue of C. Show that 12 Conclude that C has no real eigenvalues. [1] [3] =-1. [3]
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
Problem. Let A=1-1-2-2-2 0-2 1 1 -1 21 0 (a) Find a Jordan form J for A (b) Find the change of basis matrix X such that X-1 AX = J. Problem. Let A=1-1-2-2-2 0-2 1 1 -1 21 0 (a) Find a Jordan form J for A (b) Find the change of basis matrix X such that X-1 AX = J.
4. We saw in class that if A is an orthogonal matrix, then ||AX|| = ||X||. One matrix for which we know this is true is the rotation matrix, A = [cos – sin 0] sin cos a. (2 pts) Show that A is an orthogonal matrix. b. (2 pts) Since A is an orthogonal matrix, A-1 = AT. Show that AT can be written as cos 0 – sino w does the angle o relate to the angle ?...
8. Let An be the following n x n tridiagonal matrix ab 0 0 0 Cab 00 0 0 Oca 0 0 0. 0 0 0 C a Show that AnalAn- 1l-bc|A,-21 for n 2 3. If a = 1+bc, show that |An 1+bc+ (be)2 ++(bc)" If a 2 cos with 0 <0<T and b c 1 then show that sin (n+1)0 |An = sin 0 nn change 8. Let An be the following n x n tridiagonal matrix ab...
7. Given cos 20 = --and 180° <0 < 270°, find values of sino and cose.