vectors

vectors - EE103 (Fall 2009-10) 1. Vectors • definition...

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Unformatted text preview: EE103 (Fall 2009-10) 1. Vectors • definition and notation • inner products and norms • linear functions • angle between vectors 1-1 Vectors n-vector x : x = x 1 x 2 . . . x n • also written as x = ( x 1 , x 2 , . . . , x n ) • set of n-vectors is denoted R n • x i : i th element or component or entry of x Special vectors (size follows from context) • x = 0 (zero vector): x i = 0 , 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 Vectors 1-2 Vector operations Scalar multiplication of a vector x with a scalar α α x = α x 1 α x 2 . . . α x n Addition and subtraction of two n-vectors x , y x + y = x 1 + y 1 x 2 + y 2 . . . x n + y n , x − y = x 1 − y 1 x 2 − y 2 . . . x n − y n Vectors 1-3 Geometrical interpretation for n ≤ 3 : x is a point with coordinates x i Example : x = (4 , 0) , y = (2 , 2) x . 75 x y 1 . 5 y . 75 x + 1 . 5 y . 75 x = bracketleftbigg 3 bracketrightbigg , 1 . 5 y = bracketleftbigg 3 3 bracketrightbigg , . 75 x + 1 . 5 y = bracketleftbigg 6 3 bracketrightbigg Vectors 1-4 Signals a function u ( t ) of time, sampled with period T for n − 1 periods u ( t ) t T 2 T 3 T ( n − 1) T x 1 x 2 x 3 x 4 x n can be represented as an n-vector with x 1 = u (0) , x 2 = u ( T ) , . . . , x n = u (( n − 1) T ) applications in signal processing, communications, control, finance, . . ....
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This note was uploaded on 10/15/2009 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.

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vectors - EE103 (Fall 2009-10) 1. Vectors • definition...

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