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lineqs - EE236A(Fall 2007-08 Lecture 2 Linear inequalities...

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Unformatted text preview: EE236A (Fall 2007-08) Lecture 2 Linear inequalities • vectors • inner products and norms • linear equalities and hyperplanes • linear inequalities and halfspaces • polyhedra 2–1 Vectors (column) vector x ∈ R n : x = x 1 x 2 . . . x n • x i ∈ R : i th component or element of x • also written as x = ( x 1 , x 2 , . . . , x n ) some special vectors: • x = 0 ( zero vector ): x i = 0 , i = 1 , . . . , n • x = 1 : x i = 1 , i = 1 , . . . , n • x = e i ( i th basis vector or i th unit vector ): x i = 1 , x k = 0 for k negationslash = i ( n follows from context) Linear inequalities 2–2 Vector operations multiplying a vector x ∈ R n with a scalar α ∈ R : αx = αx 1 . . . αx n adding and subtracting two vectors x , y ∈ R n : x + y = x 1 + y 1 . . . x n + y n , x − y = x 1 − y 1 . . . x n − y n x . 75 x y 1 . 5 y . 75 x + 1 . 5 y Linear inequalities 2–3 Inner product x , y ∈ R n ( x, y ) := x 1 y 1 + x 2 y 2 + ··· + x n y n = x T y important properties • ( αx, y ) = α ( x, y ) • ( x + y, z ) = ( x, z ) + ( y, z ) • ( x, y ) =...
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lineqs - EE236A(Fall 2007-08 Lecture 2 Linear inequalities...

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