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lecture14

# lecture14 - Lecture 14 Simplex Hyper-Cube Convex Hull and...

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Lecture 14 Simplex, Hyper-Cube, Convex Hull and their Volumes Shang-Hua Teng

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Linear Combination and Subspaces in m-D • Linear combination of v 1 (line) { c v 1 : c is a real number} • Linear combination of v 1 and v 2 (plane) { c 1 v 1 + c 2 v 2 : c 1 ,c 2 are real numbers} • Linear combination of n vectors v 1 , v 2 ,…, v n (n Space) { c 1 v 1 +c 2 v 2 +…+ c n v n : c 1 ,c 2 ,…,c n are real numbers} Span( v , v ,…, v )
Affine Combination in m -D ( 29 ( 29 { } = + + + + + + - + + + - = 1 : 1 , , , affine 2 1 2 2 1 1 1 1 2 2 1 1 2 1 n n n n n j j n p p p p p p p p p α

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Convex Combination in m -D ( 29 = + + + + + + n i p p p p p p n n n n 1 , 0 1 : , , , Convex i 2 1 2 2 1 1 2 1 α y p 1 p 2 p 3
Simplex n dimensional simplex in m dimensions (n < m) is the set of all convex combinations of n + 1 affinely independent vectors

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Parallelogram
Parallelogram

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Hypercube n-cube { } [ ] T i n n e n i c e c e c e c c 0 1 0 1 , 1 , 0 : 0 convex i 2 2 1 1 0 = + + + + (1,0) (0,1) (1,0,0) (0,0,1) (1,1,1)
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lecture14 - Lecture 14 Simplex Hyper-Cube Convex Hull and...

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