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(a) Reduce the following matrices to diagonal form and find a g-inverse of each 120-11 4 5 6 2 2 3 -1 A=158 O 11 and B-1084 7 1o-2 3 21 6 (5+5 (b) () For any n x I vector a 0, show that a (ii) Find the g-inverse of the vector a, where a' = [1 a'a 5 2] 3 1 (a) Reduce the following matrices to diagonal form and find a g-inverse of each 120-11 4 5...
, then n lim Let Ά be a square matrix. Prove that if ρ(A)<1 Use the following fact without proof. For any square matrix A and any positive real number ε , there exists a natural matrix norm I l such that l-4 ll < ρ (d) +ε IIA" 11-0
3. For n 2 2, let X have n-dimensional normal distribution MN(i, V). For any 1 3 m < n, let X1 denote the vector consisting of the last n - m coordinates of X < n, let 1 (a). Find the mean vector and the variance-covariance matrix of X1. (b). Show that Xi is a (n- m)-dimensional normal random vector.
6. Let R be a ring, and let 11 and 12 be ideals of R. We define the product of 11 and 12 to be 1112 = {TER:r => aibi, with k > 1, Q1, ..., ak € 11, b1,..., bk € 12 In other words, an element of the product 1.12 is a finite sum of products a;bi, where a, comes from I and bi comes from 12. (a) Prove that 11 12 is an ideal of R, contained...
(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...
I am struggling with part (ii) Let g(x, y) (e" +1)2+2(e-e(e1). 22-1 For any fixed x E R, show that the equation g(x,y) = 0 admits a solution y(x) > 0, and limx-0 y(x) = 0. (ii) Show that there exists a constant y > 0, such that for any fixed y E [0, ] the equation g(x,y) = 0 admits a solution 2(y).
Q6 (4+3+3+ 6=16 marks) Let Xo, X1, X2 be three distinct real numbers. For polynomials p(x) and q(x), define < p(x),q(x) >= p(xo)q(x0) + p(x1)q(x1) + p(x2)q(22). Let p(n) denote the vector space of all polynomials with degree more no than n. (i) Show that < .. > is an inner product in P(2). (ii) Is < ... > an inner product in P(3)? Explain why. (iii) Is <,:> an inner product in P(1)? Explain why. (iv) Consider Xo =...
I need answers for question ( 7, 9, and 14 )? 294 Chapter 6. Eigenvalues and Eigenvectors Elimination produces A = LU. The eigenvalues of U are on its diagonal: they are the . The cigenvalues of L are on its diagonal: they are all . The eigenvalues of A are not the same as (a) If you know that x is an eigenvector, the way to find 2 is to (b) If you know that is an eigenvalue, the...
Please show work Consider O the matrix A= o 15 I 12 o 1 -2 14 a) Find all the eigenvalues of A. Find any b) One of A's eigenvalues is x=2 . eigenvector corresponding to a= 2.
1. Write in the product 1. 03 1-sec-buty!cyclopentene > … 2. Zn, H O 2. a) Name this compound according to the IUPAC system: CH,-CH-CHCEC-CH.C(CHa CH b) Write in the product of the compound in 2 a) with the following reagents. Include stereochemistry where relevant. excess H2, Pd/C Na, NH H2, Lindlar catalyst 1) 2) 3) 3. Write in the product of 4-methyl-1-pentyne with the following reagents 1) 2) 2 HBr 3) 4. Show how to transform the compound on...