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# hw#2 - 2 5 and a 3 =[5 2 3 5 Are these vectors linearly...

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HOMEWORK 2, OPERATIONS RESEARCH January 18, 2012. Name: Due on Monday, January 23. Write the number of the assignment at the top of the first page of your assignment. Don’t forget to write down your name. Write the complete statement of each question as well as the answer. In each question, show all your work. Answer the questions neatly. Staple all the pages. You can work cooperatively on this set of problems but write your own solutions. Textbook, page 77-81. Ex. 3.1-2, 3.1-7(a,b), 3.1-8, 3.2-3(a,b), 3.3-1. Exercise 1 Show that the vectors u 1 = [1 , 0 , 0] , u 2 = [0 , 1 , 0] , and a 3 = [1 , 5 , 3] form a basis for R 3 . Replace u 2 by [0 , 1 , 1] . Is the new set still a basis? Exercise 2 Determine if the following collections of vectors form a basis for R 3 , a generating set for R 3 , or neither. Justify your answer. • { [1 , 2 , 1] , [ - 1 , 0 , - 1] , [0 , 0 , 1] } . • { [1 , 3 , 2] , [2 , 0 , 5] } . • { [ - 1 , 2 , 3] , [0 , 1 , 0] , [1 , 2 , 3] , [ - 3 , 2 , 4] } . Exercise 3 Let u 1 = [ - 1 , 2 , 0] , u 2 = [3

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Unformatted text preview: 2 , 5] , and a 3 = [5 / 2 , 3 , 5] . Are these vectors linearly independent? Do they span R 3 ? 1 Exercise 4 Find the hyperplane H in R 4 which contains the four points e 1 , e 1 + 2 e 2 , e 2 + 3 e 3 , e 3 + 4 e 4 where { e 1 ,e 2 ,e 3 ,e 4 } is the canonical basis for R 4 . Exercise 5 Find the hyperplane in R 4 containig the four points , e 1 + e 2 , e 1-e 2 +2 e 3 , 3 e 4-e 2 . Find also the value of t for which t ( e 1-e 2 )+(1-t ) e 4 is in the hyperplane. Exercise 6 Let l denote the line in R 4 through e 1-e 3 and-e 1 + e 2 + 2 e 4 . Find the hyperplane H through e 1-e 3 to which l is perpendicular. Exercise 7 Find a linear programming model for the Wyndor Glass Co. problem. Explain what you do. Draw the feasible set for this problem as well. 2...
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hw#2 - 2 5 and a 3 =[5 2 3 5 Are these vectors linearly...

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