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Vector Space Axioms, Zero Element and Geometric Method of Linear Programming

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1) Let R+={x/0<x} (that is, the set of positive real numbers). Define the operation of addition on this set by x+y=xy. Show that with this definition there is a zero element, and that every x in R+ has an inverse. Determine what the zero element is, and for any given x, what its additive inverse is.

2) Let V= {(x,y)/y=x+2} with addition and multiplication by a scalar defined on V by:
(x,y) + (u,v)= (x+u,y+v-2)
k (x,y)= (kx,k (y-2) +2)
Check to see if the vector space axioms 4 (existence of a 0 vector) and 5 (existence of an inverse) are satisfied. If they are, show the 0 vector and the inverse, and if they are not, show why not.

3)Use the geometric method of linear programming to maximize the objective function f(x,y)=4x-3y subject to the following constraints.
x is greater than or equal to 0
x+2y is greater than or equal to 4
x+y is less than or equal to 6
2x-2y is less than or equal to 8
3x-y is less than or equal to 2

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Vector Space Axioms, Zero Element and Geometric Method of Linear Programming are investigated. The solution is detailed and well presented.

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