Math 3113-005 Quiz 6: Solving Matrices, Linear Independence, and Equations - Prof. Darryl , Quizzes of Mathematics

Download Math 3113-005 Quiz 6: Solving Matrices, Linear Independence, and Equations - Prof. Darryl and more Quizzes Mathematics in PDF only on Docsity! Mathematics 3113-005 Quiz 6 Form B April 8, 2011 Name (please print) Instructions: Give concise answers, but clearly indicate your reasoning. I. (5) Let A =        4t −1 0 2 1 −t 1 2 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) · (4t · 2− (−1) · 1) + 0 = 8t2 + t . II. (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. III. (3) Write the system x′ 1 = 8x1 + tx2 + cos(t), x ′ 2 = x2 − x3, x′3 = t+ 2tx2 − x3 in matrix form X ′ = PX + F . Do not proceed further with solving the system, just rewrite the general form X ′ = PX + F with X, P and F written as matrices with the correct dimensions and entries for this particular system.        x1 x2 x3        ′ =        8 t 0 0 1 −1 0 2t −1               x1 x2 x3        +        cos(t) 0 t        IV. (3) Write the second-order system x′′ − 2x + y = 0, y′′ + 2x − 3y = 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 = 2x1 − y1 y′ 1 = y2 y′ 2 = −2x1 + 3y1