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Unformatted text preview: AIMS Lecture Notes 2006 Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number of other important norms that are used in numerical analysis. In this section, we review the basic properties of inner products and norms. 5.1. Inner Products. Some, but not all, norms are based on inner products. The most basic example is the familiar dot product h v , w i = v w = v 1 w 1 + v 2 w 2 + + v n w n = n X i = 1 v i w i , (5 . 1) between (column) vectors v = ( v 1 , v 2 , . . . , v n ) T , w = ( w 1 , w 2 , . . . , w n ) T , lying in the Euclidean space R n . A key observation is that the dot product (5.1) is equal to the matrix product v w = v T w = ( v 1 v 2 . . . v n ) w 1 w 2 . . . w n (5 . 2) between the row vector v T and the column vector w . The key fact is that the dot product of a vector with itself, v v = v 2 1 + v 2 2 + + v 2 n , is the sum of the squares of its entries, and hence, by the classical Pythagorean Theorem, equals the square of its length; see Figure 5.1. Consequently, the Euclidean norm or length of a vector is found by taking the square root: k v k = v v = p v 2 1 + v 2 2 + + v 2 n . (5 . 3) Note that every nonzero vector v 6 = has positive Euclidean norm, k v k > 0, while only the zero vector has zero norm: k v k = 0 if and only if v = . The elementary properties of dot product and Euclidean norm serve to inspire the abstract definition of more general inner products. 3/15/06 77 c 2006 Peter J. Olver v 1 v 2 k v k v 1 v 2 v 3 k v k Figure 5.1. The Euclidean Norm in R 2 and R 3 . Definition 5.1. An inner product on the vector space R n is a pairing that takes two vectors v , w R n and produces a real number h v , w i R . The inner product is required to satisfy the following three axioms for all u , v , w V , and scalars c, d R . ( i ) Bilinearity : h c u + d v , w i = c h u , w i + d h v , w i , h u , c v + d w i = c h u , v i + d h u , w i . (5 . 4) ( ii ) Symmetry : h v , w i = h w , v i . (5 . 5) ( iii ) Positivity : h v , v i > whenever v 6 = , while h , i = 0 . (5 . 6) Given an inner product, the associated norm of a vector v V is defined as the positive square root of the inner product of the vector with itself: k v k = p h v , v i . (5 . 7) The positivity axiom implies that k v k 0 is real and non-negative, and equals 0 if and only if v = is the zero vector. Example 5.2. While certainly the most common inner product on R 2 , the dot product v w = v 1 w 1 + v 2 w 2 is by no means the only possibility. A simple example is provided by the weighted inner product h v , w i = 2 v 1 w 1 + 5 v 2 w 2 , v = v 1 v 2 , w = w 1 w 2 . (5 . 8) Let us verify that this formula does indeed define an inner product. The symmetry axiom (5.5) is immediate. Moreover, h c u + d v , w i = 2( c u 1 + d v 1 ) w 1 + 5( c u...
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