# Download e-book for kindle: Algorithms for Diophantine Equations [PhD Thesis] by de Weger B.M.M. By de Weger B.M.M.

ISBN-10: 9061963753

ISBN-13: 9789061963752

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Additional info for Algorithms for Diophantine Equations [PhD Thesis]

Example text

2 5 −13 . 1000  1 0 0 .. 1200    ..  . We reduce this matrix to    in a matrix:  3 −9 . 0  3  0 1 0 .. 500  . The prize of a cow, . 0 0 1 .. 300 . −600 −5 6 8 a sheep, and a pig is 1200, 500 and 300 coins, respectively. 67 The second measurement in the problem tells us that 4 sparrows and 1 swallow weigh as much as 1 sparrow and 5 swallows. We will immediately interpret this as 3 sparrows weighing the same as 4 swallows. The other measurement we use is that all the birds together weigh 16 liang.

1   . c   c2 1 c−2 2 c − 5c + 6     .  This system is consistent if and only if c = 2 or c = 3. Thus the vector is a linear combination if c = 2 or c = 3. 62 We need to solve the system  with augmented matrix      1 1 1  c  = x a  + y b  c2 a2 b2   1   a  a2 The matrix reduces to . 1..  1  .  0 b − a..  . 0 0.. 1.. b.. b2 ..   1   . c   c2 1 c−a (c − a)(c − b)     .  This system is consistent if and only if c = a or c = b. Thus the vector is a linear combination if c = a or c = b.

10b. 48 The fact that x1 is a solution of Ax = b means that Ax1 = b. a. A(x1 + xh ) = Ax1 + Axh = b + 0 = b b. A(x2 − x1 ) = Ax2 − Ax1 = b − b = 0 c. Parts (a) and (b) show that the solutions of Ax = b are exactly the vectors of the form x1 + xh , where xh is a solution of Ax = 0; indeed if x2 is a solution of Ax = b, we can write x2 = x1 + (x2 − x1 ), and x2 − x1 will be a solution of Ax = 0, by part (b). 14; the line L runs through the tip of x1 and is parallel to the given line consisting of the solutions of Ax = 0.