exam1d - 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: Mon 28 Sept 1998 Problem 1: (a) Using Gauss—Jordan, find the row—reduced echelon form of the following aug— mented matrix. (INDICATE the row operations you use). A?“ 1 2 3 6 AfgxgvAéxg 1 0 —1 0 —> 0 1 2 3 —> 0 1 2 3 0 —1 —2 —3 0 0 0 0 (b) Then give the solutions of the corresponding linear system A50 : I). So solutions are (7‘, 3 — 27‘, 7‘) for free variable x3 = 7‘. Problem 2: (a) What elementary row OPERATION Will change 12 1 2 _ _ 7 A (2 1) toB (0 _3). A?“ : add —2 times 1st row to 2nd. (b) What elementary row MATRIX E will, by left muliplication, perform the same operation? (that is, EA = B) 1 . . . . . . E = < _2 (1) > : (obtained by domg that operation to the 1dent1ty matrix). Problem 3: (a) Find the inverse (any method) of: 1 0 0 A : ( 1 1 0 ) . I I 1 Using (All) method: use A§1X1,A§1X1 to clear first column; then A§1X2 to clear second column. 1 0 0 Get inverse: —1 1 0 0 —1 0 (b) Give the LU —decomposition of 12 A—(m) that is, find lower-triangular L and upper-triangular U, so that A = LU. Apply 142—1“ to get U : < (1) i 80 L from inverse operation A31“ is ( 1 [1) > Problem 4: (a) Find the determinant of Top row: 1(1.1 — 0.1) — 1(0.1 — 1.1) + 0 = (1) — (—1) = 2. (b) Use Cramer’s rule to solve 1 2 4 3 4 1 10 det(A) = 1.4 — 3.2 = —2, so 1161 = —§(4.4 — 10.2): 2 and $2 = — Her—1 Ol—‘H 1—11—10 (1.10 — 3.4) = 1. 1 2 Problem 5: (a) Is (1,0,1) in the span of (1,1,1) and (1,2,1)? Either give coefficients in a linear combination, or explain Why it is not possible. 1 1 1 ROW-reduction quickly produces coeflicients 2 and —1. (b) Let V be the space of 2 X 2 matrices, and W the subSET of diagonal matrices. Show that W is a subSPACE of V. A E W says diagonal, meaning 0 of the diagonal, so A has form < a 0 1 1 1 Yes: Set up augmented matrix (A|b) : ( 1 2 0 ) . 0 b 0 d a+c 0 0 b+d Similarly ifB E W it has form < C 0 > Is A + B E W? It is < ), also diagonal, so yes. m 0 0 Tb We see “yes”, W is a subspace of V. For scalar 7‘, is TA E W? It is < >, also diagonal, so yes. ...
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exam1d - Math 310: Hour Exam 1 (Solutions) Prof. S. Smith:...

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