N ex 1 the euclidean inner product for rn show that

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5.28nEx 1: The Euclidean inner product for RnShow that the dot product in Rnsatisfies the four axioms of an inner productSol:Thus, the dot product can be a kind of inner product in Rnnnvuvuvu+++==L2211,vuvu??),,,(,),,,(2121nnvvvuuuLL==vu
5.29nEx 2: A different inner product for RnShow that the following function defines an inner product on R2. Given , w
5.30 n Note: Example 2 can be generalized such that can be an inner product on Rn 0 , , 2 2 2 1 1 1 + + + = i n n n c v u c v u c v u c L v u 1 1 2 2 1 1 2 2 (3) , ( 2 ) ( ) 2( ) , c c u v u v cu v cu v c = + = + = u v u v 2 2 1 2 (4) , 2 0 v v = + v v 2 2 1 2 1 2 , 0 2 0 0 ( ) v v v v = + = = = = v v v 0
3 R 3
5.32 n Ex 6: An inner product in the polynomial space is an inner product Sol: and 2 2 2 Let ( ) 1 2 , ( ) 4 2 be polynomials in p x x q x x x P = - = - + 0 0 1 1 , n n p q a b a b a b + + + L (a) , ? p q = (b) || || ? q = (c) ( , ) ? d p q = (a) , (1)(4) (0)( 2) ( 2)(1) 2 p q = + - + - = 2 2 2 (b) || || , 4 ( 2) 1 21 q q q = = + - + = 2 2 2 2 (c) 3 2 3 ( , ) || || , ( 3) 2 ( 3) 22 p q x x d p q p q p q p q - = - + - = - = - - = - + + - = Q , and For 1 0 1 0 n n n n x b x b b q x a x a a p + + + = + + + = L L
5.33 n Theorem 5.7: Properties of inner products Let u, v , and w be vectors in an inner product space V , and let c be any real number (1) (2) (3) To prove these properties, you can use only the four axioms in the definition of inner product (see Slide 5.26) Pf: (1) (2) (3) , , 0 = = 0 v v 0 , , , + = + u v w u w v w 長 長 , , c c = u v u v 0 , 0 , , 0 = = = u v u v 0 v , , , , , , + = + = = u v w w u v w u w v u w v w 長 +長 長 +長 , , , c c c = = u v v u u v
: :
5.35 n Normalizing vectors (1) If , then v is called a unit vector (2) (the unit vector in the direction of v ) (if v is not a zero vector) 1 || || = v v 0 Normalizing → v v
5.36 n Properties of norm: (the same as the properties for the dot product in Rn on Slide 5.2) (1) (2) if and only if (3) n Properties of distance: (the same as the properties for the dot product in Rn on Slide 5.9) (1) (2) if and only if (3) 0 || || u 0 || || = u 0 u = || || | | || || u u c c = 0 ) , ( v u d 0 ) , ( = v u d v u = ) , ( ) , ( u v v u d d =
5.37 n Theorem 5.8 長 Let u and v be vectors in an inner product space V (1) Cauchy-Schwarz inequality: (2) Triangle inequality: (3) Pythagorean theorem: u and v are orthogonal if and only if Theorem 5.5 Theorem 5.6 Theorem 5.4 || || || || || || v u v u + + 2 2 2 || || || || || || v u v u + = + || || || || | , | v u v u

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