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practiceExam1 - Math 310 Hour Exam 1(Solutions Prof S Smith...

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Unformatted text preview: Math 310: Hour Exam 1 (Solutions) Prof. S. Smith: Fall 1995 Problem 1: (a) Find the row-reduced echelon form of A11 1 1 2 3 1 2 3 . —4x1 —7x1 M_1X. —2x2 6x2 A2 4‘3 0 —3 —6 —3>2 0 1 2 A1 4A3 0 —6 —12 0 —6 —12 (b) What are the solutions of the system An = 0 ? (Check!) Third variable is free, so solutions 503(1, —2, 1)T. Problem 2: Give the LU —decomposition of 11 A-(ls) that is7 find lower-triangular L and upper-triangular U, so that A : LU. A271“ 1 1 . . 1 1 1 0 Get U from —> 0 2 . So L from 1nverse operat1on A; X = 1 1 QUOTE? «3mm OOH Problem 3: Use Cramer’s rule (determinants) to solve Ax = b given by We 11:). . 1 1 _ _ l 5 1 _ _ 1 5 _ Flrstdet<1 2>_1,80951—(1)det<7 2)—3and$2—det<1 7)—2. Problem 4: (a) Find the inverse (by any method) of 12(13). . . . . 1 2 . . 5 —2 —5 2 Quick Via adJomt. det ( 3 5 > — —1,s01nverse 1s — < _3 1 > — < 3 _1 (b) Use the above to express the solutions of Ax = b in terms of the constants b1 and b2. By A_1b, namely $1 = —5b1 + 2b2 and 332 = 3b1 — b2. Problem 5: (a) Is (1, 2,3) in the span of (4,0, 5) and (6,0, 7) ? Noifor example, any linear combination of the latter two vectors has 0 in second entry. (b) Let V be the space of all functions (with at least 2 derivatives). Let W be the subSET af all functions f which are Salutiens ef the differential equatien f”+5f=0. Shaw that the Salutien set W is a subSPACE ef V. Take f,g E W, and scalar c: Then we have f” + 5f 2 0 = g” + 59. Isf+g€ W? (f——g)”+5(f+g) = (f”+5f)+(g”+5g) =0+0=0, so OK. Is cf 6 W? (cf)” —— 5(cf) : c(f” + 5f) 2 0(0) 2 0, also OK. ...
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