04-07-03 - Lecture 25 Andrei Antonenko April 7, 2003 1...

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Lecture 25 Andrei Antonenko April 7, 2003 1 Euclidean spaces Definition 1.1. Let V be a vector space. Suppose to any 2 vectors v,u V there assigned a number from R which will be denoted by h v,u i such that the following 3 properties hold: Bilinearity • h au 1 + bu 2 ,v i = a h u 1 ,v i + b h u 2 ,v i • h u,av 1 + bv 2 i = a h u,v 1 i + b h u,v 2 i Reflexivity h u,v i = h v,u i Positivity h u,u i ≥ 0 ; moreover if h u,u i = 0 , then u = 0 . Any function which satisfy properties above is called a scalar (inner) product . A vector space V with a scalar product is called a (real) Euclidean space . Now we will give popular examples of the scalar products in different spaces. R n The scalar product of 2 vectors x and y from R n can be defined as following: if x = ( x 1 ,x 2 ,...,x n ) and y = ( y 1 ,y 2 ,...,y n ) then h x,y i = x 1 y 2 + x 2 y 2 + ··· + x n y n . If the vectors are represented by column-vectors, i.e. by n × 1-matrices, then h x,y i = x > y. Example 1.2. Let x = (1 , 2 , 3) and y = (3 , - 1 , 4) . Then h x,y i = 1 · 3+2 · ( - 1)+3 · 4 = 13 . Actually, there are other ways of defining a scalar product in the vector space R n . For example, given positive numbers a 1 ,a 2 ,...,a n we can define the scalar product to be h ( x 1 ,x 2 ,...,x n ) , ( y 1 ,y 2 ,...,y n ) i = a 1 ( x 1 y 1 ) +
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04-07-03 - Lecture 25 Andrei Antonenko April 7, 2003 1...

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