705 - u v = v u when the field in discussion is C Property...

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1 u , v = v , u when the field in discussion is C . Property: Many inner products may be defined on a vector space. Proof If ·, · is an inner product, then ·, · r defined by u , v r = r u , v is also an inner producr for any r > 0. Definition. A vector space endowed with a particular inner product is called an inner product space . Example: The dot product is an inner product on R n . Example: V = C ([ a , b ]) = { f | f : [ a , b ] R , f is continuous} is a vector space, and the function ·, · : V × V R defined by f , g V is an inner product on V . Axiom 1: f 2 is continuous and non-negative. f 0 f 2 ( t 0 ) > 0 for some t 0 [ a , b ]. f 2 ( t ) > p > 0 [ t 0 r /2, t 0 + r /2] [ a , b ]. Axioms 2 - 4: You examine them. = b a dt t g t f g f ) ( ) ( , . 0 ) ( , 2 > = p r dt t f f f b a Example: A , B = trace( AB T ) is the Frobenius inner product on R n × n . Axiom 1: A , A = trace( AA T ) = Σ 1 i , j n ( a ij ) 2 > 0 A O . Axiom 2: trace( AB T ) = trace( AB T ) T = trace( BA T ). Axioms 3 - 4: You examine them.
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2 Definitions. For any vector v in an inner product space V , the norm or length of v is denoted and defined as || v || = v , v 1/2 . The
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705 - u v = v u when the field in discussion is C Property...

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