By Heinrich Dorrie
Difficulties that beset Archimedes, Newton, Euler, Cauchy, Gauss, Monge and different greats, able to problem today's would-be challenge solvers. between them: How is a sundial developed? how will you calculate the logarithm of a given quantity with no using logarithm desk? No complicated math is needed. comprises a hundred issues of proofs.
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Extra resources for 100 Great Problems of Elementary Mathematics (Dover Books on Mathematics)
23. a. True. See the first paragraph of the subsection titled Homogeneous Linear Systems. b. False. The equation Ax = 0 gives an implicit description of its solution set. See the subsection entitled Parametric Vector Form. c. False. The equation Ax = 0 always has the trivial solution. The box before Example 1 uses the word nontrivial instead of trivial. d. False. The line goes through p parallel to v. See the paragraph that precedes Fig. 5. e. False. The solution set could be empty! The statement (from Theorem 6) is true only when there exists a vector p such that Ap = b.
So suitable values for −3 −9 3 x1 and x2 would be 3 and –1 respectively. ) Thus x = −1 satisfies Ax = 0. 34. Inspect how the columns a1 and a2 of A are related. The second column is –3/2 times the first. Put 3 another way, 3a1 + 2a2 = 0. Thus satisfies Ax = 0. 2 Note: Exercises 33 and 34 set the stage for the concept of linear dependence. 35. Look for A = [a1 a2 a3] such that 1·a1 + 1a2 + 1·a3 = 0. That is, construct A so that each row sum (the sum of the entries in a row) is zero.
0 0 In particular, x4 is free (and not zero as some may assume). The solution is x1 = 4x2 – 5x6, x3 = x6, x5 = 4x6, with x2, x4, and x6 free. In parametric vector form, x1 4 x2 − 5 x6 4 x2 0 −5 x6 0 −5 4 x 0 x2 2 x2 0 0 0 1 x3 0 0 x6 0 1 0 x6 x= = = = x2 + x4 + x6 + + x4 x4 0 x4 0 1 0 0 x5 4 x6 0 0 4 x6 0 4 0 x6 0 1 0 x6 0 0 x6 ↑ ↑ ↑ u v w Note: The Study Guide discusses two mistakes that students often make on this type of problem.
100 Great Problems of Elementary Mathematics (Dover Books on Mathematics) by Heinrich Dorrie