Final Exam Solutions - Matrix Analysis | MATH 410, Exams of Mathematics

Download Final Exam Solutions - Matrix Analysis | MATH 410 and more Exams Mathematics in PDF only on Docsity! 6 te 2) - ede 50 Od _ EY | 3K? i! (i ff oa VY (| / | Math 410 (Prof. Bayly/Foth) FINAL EXAM: Wednesday 10 August 2005 j There are 10 problems on this exam. They are not all the same length or difficulty, nor the 4 | { y ° i same number of points. You should read through the entire exam before deciding which { problems you will work on earlier or later. You are not expected to complete everything, ? j but you should do as much as you can. You will not need a calculator on this exam. If your calculations become numerically awkward and time-consuming, you may describe the steps you would take if you had a calculator. It is EXTREMELY important to show your work! Correct answers without documented support will have points deducted. Vin. torr clie dy Nix pron by Milerinnitonts , (1})(10 points) For what values of k is the quadratic form 3x* + ky? — 8ay+az+ 2° positive definite? bye g(x) 2 KTAK wow J. [3-4 % Yk og % Oo | T; All [e- Lif | [Lave — PERDUE c U2 7% Aveh (3-4 %e \ iM. a -ib 2 PT a O ka 2 a elt NO XN o - > > a (2)(10 points) Find the closest point on the plane spanned by vectors vy = (1,0, 1)? and vo = (0,1,2)* to the point P = (3, 2,1). Also compute the shortest: distance. Ja. Ed wy so Zot r(: Ly! g ) Te. Ty He adn [! IC / p Yt (2 2G apy) yoy. ; 29 “>| he i +0 {c (2 wee s S » AK = My sh — PAN ad ~~ & (4)(20 points) Consider the symmetric matrix A = C 5): (a)(10 points) Find the eigenvalues and eigenvectors of A, and verify that the cigenvectors are orthogonal. (b)(10 points) Find a matrix Q whose columns are orthonormal vectors, for which QT AQ = A, a diagonal matrix. Verify by direct calculation of Q7 AQ. © Eveline bet (4-00) 20 hit (AAT) = hiA oA ‘) 2 b-A (b) (Lime GQ Ar Lure NanMhu ze Cie reos a at tMtiwnr fe Ge Fok (ip G/L (2 HUNTS EES) <P Yea! (5)(10 points) The Goose and Gherkin and No Octopi are neighboring restaurants that start the year with 75 customers each. ‘Phe Goose regularly presents live music, with the result that 80 per cent of the patrons one night return on the next night, with the other 20 per cent going to No Octopi for some quiet pizza. Meanwhile 60 per cent of the customers at. No Octopi return the next night, with 40 per cent going over to the Goose. As weeks and weeks go by (i.e. as time goes to infinity), what are the expected numbers of customers at the two restaurants? —T of, . a a lv \ ¢ 2 ae / Hitwe. pins ows a | te the § sayrt Cig ja He Lave Y a ted TOTAL pueden = Tiled ol: cy prevtn (G0 ayI90 > GO x= Dy (VU / 0 COU = Expecr |? NW Cate } He (eo coco a MW Ocifl (7)(10 points) Construct polynomials Po, P,, and P, of degree 0, 1, and 2 respectively, which are orthogonal with respect to the inner product | ‘ F@Og(t de. Lets fet with Mgoy f(t) - 4 f= t , Rls & (ram Shinde PoC = P(e p.)= f= b, g> ph (+) po, fo? co MD - y (t-%) ie td = jee +4) tat - [aoe f (3 Le44¢) dt = “4B 2 y2 be bh 4 2 [+ F+gellsd TG = 36% ol, Li, h>= j e) )O tee | eeu [legate Yt peg ae Hor poh? feed. eu = PAL eM W y(t: t- V4 (1)