Quiz 6 in Mathematics 3113-005: Linear Independence, Determinants, and System of Equations, Quizzes of Mathematics

Download Quiz 6 in Mathematics 3113-005: Linear Independence, Determinants, and System of Equations and more Quizzes Mathematics in PDF only on Docsity! Mathematics 3113-005 Quiz 6 Form A April 8, 2011 Name (please print) Instructions: Give concise answers, but clearly indicate your reasoning. I. (2) Define what it means to say that a collection of vectors {X1,X2, . . . ,Xn} is linearly independent. It means that c1X1 + c2X2 + · · ·+ cnXn = 0 for constants ci only when all the ci are 0. [or] It means that if c1X1 + c2X2 + · · ·+ cnXn = 0 for constants ci, then all the ci are 0. II. (5) Let A =        3t −1 0 2 1 −t 1 5 0        , B =     3 −1 0 2− t 1 1     , and C = [ − cos(t) 3 0 ] . (a) Tell which of the following six products are defined (do not do any calculations, just tell which ones are defined): AB, BA, AC, CA, BC, CB. BA and CA are defined, AB, AC, BC, and CB are not defined. (b) Calculate det(A). The easiest way is to expand down the third column of A, since that column contains two zeros. We find that det(A) = 0− (−t) · (3t · 5− (−1) · 1) + 0 = 15t2 + t . III. (3) Write the second-order system x′′ − 5x + 3y = 0, y′′ + 2x + y = 0 as an equivalent system of first-order equations. We define new functions x1, x2, y1, and y2 by x1 = x, x2 = x ′, y1 = y, and y2 = y ′. With these definitions, the second-order system becomes x′ 1 = x2 x′2 = 5x1 − 3y1 y′1 = y2 y′2 = −2x1 − y1