6.1 - MATH1850U/2050U: Chapter 6 1 INNER PRODUCT SPACES...

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MATH1850U/2050U: Chapter 6 1 INNER PRODUCT SPACES Inner Products (Section 6.1; pg. 335) Definition: An inner product on a real vector space V is a function that associates a real number v u , with each pair of vectors u and v in V in such a way that the following axioms are satisfied for all vectors u , v , and w in V and all scalars k . 1) u v v u , , (symmetry axiom) 2) w v v u w v u , , , (additivity axiom) 3) v u v u , , k k (homogeneity axiom) 4) 0 , v v and 0 , v v iff 0 v A real vector space with an inner product is called a real inner product space . Note: The dot product is not the only inner product that you can define on R n . Any rule that satisfies the definition of inner product can be used. Definition: For example, an alternative inner product on R n is the weighted Euclidean inner product with non-negative weights n w w w , , , 2 1 which is defined by the formula n n n v u w v u w v u w 2 2 2 1 1 1 , v u where ) , , , ( 2
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6.1 - MATH1850U/2050U: Chapter 6 1 INNER PRODUCT SPACES...

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