Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.1

Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.1

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9.1 Systems of Linear Inequalities 9.2 Linear Programming Involving Two Variables 9.3 The Simplex Method: Maximization 9.4 The Simplex Method: Minimization 9.5 The Simplex Method: Mixed Constraints LINEAR PROGRAMMING 9 514 ohn von Neumann was born in Budapest, Hungary, where his father was a successful banker. John’s genius was recog- nized at an early age. By the age of ten, his mathematical knowledge was so great that instead of attending regular classes he studied privately under the direction of leading Hungarian mathematicians. At the age of twenty-one, he acquired two degrees, one in chemical engineering at Zurich, and the other a Ph.D. in mathematics from the University of Budapest. He spent some time teaching at the University of Berlin, and then, in 1930, accepted a visiting professorship at Princeton University. In 1933, John von Neumann and Albert Einstein were among the first full professors to be appointed to the newly organized Institute for Advanced Study at Princeton. During World War II, John von Neumann was a consultant at Los Alamos, and his research helped in the development of the atomic bomb. In 1954, President Eisenhower appointed him to the Atomic Energy Commission. John von Neumann is considered to be the father of modern game theory––a branch of mathematics that deals with strategies and de- cision making. Much of his results concerning game theory were published in a lengthy paper in 1944 titled Theory of Games and Economic Behavior ,written with Oskar Morgenstern. In 1955, John von Neumann was diagnosed with cancer––he died in 1957 at the age of 53. Many stories are told of his mental abilities. Even during the final months of his life, as his brother read to him in German from Goethe’s Faust, each time a page was turned John would recite from memory the continuation of the passage on the following page. 9.1 SYSTEMS OF LINEAR INEQUALITIES The following statements are inequalities in two variables. and An ordered pair is a solution of an inequality in x and y if the inequality is true when a and b are substituted for x and y ,respectively. For instance, is a solution of the inequality because The graph of an inequality is the collection of all solutions of the inequality. To sketch the graph of an inequality such as begin by sketching the graph of the corresponding equation 3 x 2 2 y 5 6. 3 x 2 2 y < 6, 3 s 1 d 2 2 s 1 d 5 1 < 6. 3 x 2 2 y < 6 s 1, 1 d s a , b d x 1 y 6 3 x 2 2 y < 6 John von Neumann 1903–1957 J
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The graph of the equation will normally separate the plane into two or more regions. In each such region, one of the following must be true. (1) All points in the region are solutions of the inequality. (2) No points in the region are solutions of the inequality. So, you can determine whether the points in an entire region satisfy the inequality by simply testing one point in the region.
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This note was uploaded on 02/24/2012 for the course MATH 310 taught by Professor Staff during the Spring '08 term at VCU.

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Elementary Linear Algebra 6e - Larson, Edwards, Falvo - Chapter 9.1

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