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lecture03_mathematical notations

lecture03_mathematical notations - Yinyu Ye MS&E Stanford...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 1 Mathematical Notations Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 2 Real n -Space; Euclidean Space R : real numbers R n : n -dimensional Euclidean space x y means x j y j for j = 1 , 2 , ..., n 0 : vector of all zeros; e : vector of all ones Inner-product of two vectors: x y := x T y = n X j =1 x j y j Euclidean norm : 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

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 3 Column vector : x = ( x 1 ; x 2 ; . . . ; x n ) and row vector : x = ( x 1 , x 2 , . . . , x n ) Transpose operation : A T A set of vectors a 1 , ..., a m is said to be linearly dependent if there are scalars λ 1 , ..., λ m , not all zero, such that the linear combination m X i =1 λ i a i = 0 A linearly independent set of vectors that span R n is a basis . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 4 Plane and Half-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 0 (5,0) (0,3) Figure 1: Plane and Half-Spaces Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 6 System of Linear Equations Solve for x ∈ R n from: a 1 x = b 1 a 2 x = b 2 · · · · · a m x = b m A x = b Basic solution : select m columns from A to form a square matrix A B such that A B x B = b , the rest of x N = 0 where B is the index set of selected m coolumns.

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 7 x 0 3x+2y=12 2x+3y=12 (2.4,2.4) (4,0) (6,0) (0,4) y (0,6) Figure 2: System of Linear Equations Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 8 Gaussian elimination method a 11 A 1 . 0 A 0 x 1 x 0 = b 1 b 0 . A = L U C 0 0
Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #03 9 Fundamental theorem of linear equations Theorem 1 Let A ∈ R m × n and b ∈ R m . The system { x : A x = b } has a solution if and only if that A T y = 0 and b T y 6 = 0 has no solution.

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lecture03_mathematical notations - Yinyu Ye MS&E Stanford...

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