lecture03_mathematical notations

lecture03_mathematical notations - Yinyu Ye,...

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Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 1 MathematicalNotations YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 2 Real n-Space;EuclideanSpace R :realnumbers R n : n-dimensional Euclideanspace x y means x j y j for j =1 , 2 ,...,n :vectorofallzeros; e :vectorofallones Inner-product oftwovectors: x y := x T y = n X j =1 x j y j Euclideannorm : k x k 2 = x T x , Infinity-norm : k x k =max {| x 1 | , | x 2 | ,..., | x n |} , p-norm : k x k p = n j =1 | x j | p 1 /p Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 3 Columnvector : x =( x 1 ; x 2 ; ... ; x n ) and rowvector : x =( x 1 ,x 2 ,...,x n ) Transposeoperation : A T Asetofvectors a 1 ,..., a m issaidtobe linearlydependent ifthereare scalars 1 ,..., m ,notallzero,suchthatthe linearcombination m X i =1 i a i = A linearlyindependent setofvectorsthatspan R n isa basis . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 4 PlaneandHalf-Spaces H = { x : ax = n X j =1 a j x j = b } H + = { x : ax = n X j =1 a j x j b } H- = { x : ax = n X j =1 a j x j b } Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 5 x y 3x+5y>15 3x+5y<15 3x+5y=15 (5,0) (0,3) Figure1:PlaneandHalf-Spaces Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 6 SystemofLinearEquations Solvefor x R n from: a 1 x = b 1 a 2 x = b 2 a m x = b m A x = b Basicsolution :select m columnsfrom A toformasquarematrix A B suchthat A B x B = b , therestof x N = where B istheindexsetofselected m coolumns. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 7 x 3x+2y=12 2x+3y=12 (2.4,2.4) (4,0) (6,0) (0,4) y (0,6) Figure2:SystemofLinearEquations Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 8 Gaussianeliminationmethod a 11 A 1 . A x 1 x = b 1 b . A = L U C Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 9 Fundamentaltheoremoflinearequations...
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lecture03_mathematical notations - Yinyu Ye,...

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