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A practical approach to linear algebra by Choudhary P.

By Choudhary P.

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B) if el , e2 are convex cones, their intersection el ne 2 IS Xl eel' X2 ee 2 } = {XIXeel and Xce 2 } a convex cone. (c) if e is a convex cone, the dual cone e* = {Y I X'y ~ 0 for all Xee} is a convex cone. Note that e* consists of all vectors making a non-acute angle ( ~ 7r/2) with all vectors of e. Relative to fundamental operation (c) above we may take the negative of the dual cone e* of e in R n so as to form the polar cone e+ = {Y I X'y ~ 0 for all Xce} consisting of all vectors making a non-obtuse angle ( ~ 7r/2) with all vectors of e.

Looking to Farkas' system (I) of the preceding paragraph, given any feasible solution X ~ 0 to AX = b, under what circumstances can we find at least one basic feasible solution? 32. EXISTENCE THEOREM FOR BASIC FEASIBLE SOLUTIONS TO NONHOMOGENEOUS LINEAR EQUALITIES. For an (m x n) matrix A with p(A) = m AX = b, X ~

The next theorem considers a key property of the set of all convex combinations of a finite number of points. 19. THEOREM. Let {Xl"'" Xm} be a finite collection of points in Rn. Then the set of all convex combinations of the Xi' i = 1, ... , m, m m g; = {XIX = AiXi' = 1, 0 ~ AitR for all i=1 i=1 is convex. L L\ i} We note briefly that if 'JI,. ,'J m are each convex sets in R n , then their Convex combination 'J = E~l Ai 'J i , E~l Ai = 1, 0 ~ \ t R is also convex. 20. THEOREM. A set

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