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Unformatted text preview: EE236A (Fall 200708) Lecture 3 Geometry of linear programming • subspaces and affine sets, independent vectors • matrices, range and nullspace, rank, inverse • polyhedron in inequality form • extreme points • the optimal set of a linear program 3–1 Subspaces S ⊆ R n ( S negationslash = ∅ ) is called a subspace if x, y ∈ S , α, β ∈ R = ⇒ αx + βy ∈ S αx + βy is called a linear combination of x and y examples (in R n ) • S = R n , S = { } • S = { αv  α ∈ R } where v ∈ R n ( i.e. , a line through the origin) • S = span( v 1 , v 2 , . . . , v k ) = { α 1 v 1 + ··· + α k v k  α i ∈ R } , where v i ∈ R n • set of vectors orthogonal to given vectors v 1 , . . . , v k : S = { x ∈ R n  v T 1 x = 0 , . . . , v T k x = 0 } Geometry of linear programming 3–2 Independent vectors vectors v 1 , v 2 , . . . , v k are independent if and only if α 1 v 1 + α 2 v 2 + ··· + α k v k = 0 = ⇒ α 1 = α 2 = ··· = 0 some equivalent conditions: • coefficients of α 1 v 1 + α 2 v 2 + ··· + α k v k are uniquely determined, i.e. , α 1 v 1 + α 2 v 2 + ··· + α k v k = β 1 v 1 + β 2 v 2 + ··· + β k v k implies α 1 = β 1 , α 2 = β 2 , . . . , α k = β k • no vector v i can be expressed as a linear combination of the other vectors v 1 , . . . , v i 1 , v i +1 , . . . , v k Geometry of linear programming 3–3 Basis and dimension { v 1 , v 2 , . . . , v k } is a basis for a subspace S if • v 1 , v 2 , . . . , v k span S , i.e. , S = span( v 1 , v 2 , . . . , v k ) • v 1 , v 2 , . . . , v k are independent equivalently: every v ∈ S can be uniquely expressed as v = α 1 v 1 + ··· + α k v k fact: for a given subspace S , the number of vectors in any basis is the same, and is called the dimension of S , denoted dim S Geometry of linear programming 3–4 Affine sets V ⊆ R n ( V negationslash = ∅ ) is called an affine set if x, y ∈ V , α + β = 1 = ⇒ αx + βy ∈ V αx + βy is called an affine combination of x and y examples (in R n ) • subspaces • V = b + S = { x + b  x ∈ S} where S is a subspace • V = { α 1 v 1 + ··· + α k v k  α i ∈ R , ∑ i α i = 1 } • V = { x  v T 1 x = b 1 , . . . , v T k x = b k } (if V negationslash = ∅ ) every affine set V can be written as V = x + S where x ∈ R n , S a subspace ( e.g. , can take any x ∈ V , S = V − x ) dim( V − x ) is called the dimension of V Geometry of linear programming 3–5 Matrices A = a 11 a 12 ··· a 1 n a 21 a 22 ··· a 2 n ....
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This note was uploaded on 09/26/2009 for the course CAAM 236 taught by Professor Dr.vandenber during the Spring '07 term at Monmouth IL.
 Spring '07
 Dr.Vandenber

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