mat 2310 0607_2_note9_2

mat 2310 0607_2_note9_2 - Lecture Note 9-2: March 21, 2007...

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Lecture Note 9-2: March 21, 2007 Dr. Jeff Chak-Fu WONG Department of Mathematics Chinese University of Hong Kong jwong@math.cuhk.edu.hk MAT 2310C Linear Algebra and Its Applications Spring, 2007 Produced by Jeff Chak-Fu WONG 1
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I NNER P RODUCT S PACES In this lecture we use the properties of the standard inner product or dot product on R 3 listed in Theorem 0.1 (cf. Lecture Note 7) as our foundation for generalizing the notion of the inner product to any real vector space . Hence V is an arbitrary real vector space , not necessarily finite-dimensional. I NNER P RODUCT S PACES 2
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Definition : If V be any real vector space . An inner product on V is a function that assigns to each ordered pair of vectors u , v in V a real number denoted by h u , v i satisfying (a) h u , u i ≥ 0 ; h u , u i = 0 if and only if u = 0 V , where 0 V is the zero vector in V . ( Positive definiteness ) (b) h u , v i = h v , u i for any u , v in V ( Symmetry ) (c) h u + v , w i = h u , w i + h v , w i for any u , v , w in V ( Additivity ) (d) h c u , v i = c h u , v i , for u , v in V and c a real scalar ( Homogeneity ) From these properties, it follows that • h u , c v i = c h u , v i because h u , c v i = h c v , u i = c h v , u i = c h u , v i . • h u , v + w i = h u , v i + h u , w i . I NNER P RODUCT S PACES 3
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Example 1 The dot product on R n , as defined in Lecture Note 7 is h u , v i = u · v = u 1 v 1 + u 2 v 2 + ··· + u n v n , where u = ( u 1 ,u 2 , ··· ,u n ) and v = ( v 1 ,v 2 , ··· ,v n ) is an inner product. This inner product will be called the standard inner product on R n . I NNER P RODUCT S PACES 4
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Example 2 Let V be any finite-dimensional vector space and S = { u 1 , u 2 , ··· , u n } be a basis for V . If v = a 1 u 1 + a 2 u 2 + ··· + a n u n , and w = b 1 u 1 + b 2 u 2 + ··· + b n u n , we define h v , w i = h [ v ] S , [ w ] S i = a 1 b 1 + a 2 b 2 + ··· + a n b n . It is not difficult to verify that this defines an inner product on V . This definition of h v , w i as an inner product on V uses the standard inner product on R n . Example 2 shows that we can define an inner product on any finite-dimensional vector space. Of course, it we change the basis for V in Example 2, we obtain a different inner product. I NNER P RODUCT S PACES 5
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Example 3 Let u = ( u 1 ,u 2 ) and v = ( v 1 ,v 2 ) be vectors in R 2 . We define h u , v i = u 1 v 1 - u 2 v 1 - u 1 v 2 + 3 u 2 v 2 . Show that this gives an inner product on R 2 . Solution We have h u , u i = u 2 1 - 2 u 1 u 2 + 3 u 2 2 = u 2 1 - 2 u 1 u 2 + u 2 2 + 2 u 2 2 = ( u 1 - u 2 ) 2 + 2 u 2 2 0 . Moreover if h u , u i = 0 , then u 1 = u 2 and u 2 = 0 , so u = 0 . Conversely, if u = 0 . We can also verify the remaining three properties of the preceding definition. This inner product is, of course, not the standard inner product on R 2 . Example 3 shows that on one vector space we may have more than one inner
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mat 2310 0607_2_note9_2 - Lecture Note 9-2: March 21, 2007...

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