polyhedra - EE236A(Fall 2007-08 Lecture 3 Geometry of...

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EE236A (Fall 2007-08) 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 = { 0 } • 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
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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
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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 0 + S where x 0 R n , S a subspace ( e.g. , can take any x 0 ∈ V , S = V − x 0 ) dim( V − x 0 ) 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 . . . . . . . . . a m 1 a m 2 · · · a mn R m × n some special matrices: A = 0 (zero matrix): a ij = 0 A = I (identity matrix): m = n and A ii = 1 for i = 1 , . . . , n , A ij = 0 for i negationslash = j A = diag ( x ) where x R n (diagonal matrix): m = n and A = x 1 0 · · · 0 0 x 2 · · · 0 .
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