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Unformatted text preview: Orthogonal Complements (Revised Version) Math 108A: May 19, 2010 John Douglas Moore 1 The dot product You will recall that the dot product was discussed in earlier calculus courses. If x = ( x 1 ....,x n ) and y = ( y 1 ....,y n ) are elements of R n , we define their dot product by x · y = x 1 y 1 + ··· + x n y n . The dot product satisfies several key axioms: 1. it is symmetric: x · y = y · x ; 2. it is bilinear: ( a x + x ) · y = a ( x · y ) + x · y ; 3. and it is positivedefinite: x · x ≥ 0 and x · x = 0 if and only if x = . The dot product is an example of an inner product on the vector space V = R n over R ; inner products will be treated thoroughly in Chapter 6 of [1]. Recall that the length of an element x ∈ R n is defined by  x  = √ x · x . Note that the length of an element x ∈ R n is always nonnegative. CauchySchwarz Theorem. If x 6 = and y 6 = , then 1 ≤ x · y  x  y  ≤ 1 . (1) Sketch of proof: If v is any element of R n , then v · v ≥ 0. Hence ( x ( y · y ) y ( x · y )) · ( x ( y · y ) y ( x · y )) ≥ . Expanding using the axioms for dot product yields ( x · x )( y · y ) 2 2( x · y ) 2 ( y · y ) + ( x · y ) 2 ( y · y ) ≥ or ( x · x )( y · y ) 2 ≥ ( x · y ) 2 ( y · y ) . 1 Dividing by y · y , we obtain  x  2  y  2 ≥ ( x · y ) 2 or ( x · y ) 2  x  2  y  2 ≤ 1 , and (1) follows by taking the square root. The key point of the CauchySchwarz Inequality (1) is that it allows us to define angles between vectors x and y in R n . (It is a first step towards extending geometry from R 2 and R 3 to R n .) It follows from properties of the cosine function that given a number t ∈ [ 1 , 1], there is a unique angle θ such that θ ∈ [0 ,π ] and cos θ = t. Thus we can define the angle between two nonzero vectors x and y in R n by requiring that θ ∈ [0 ,π ] and cos θ = x · y  x  y  . Then the dot product satisfies the formula x · y =  x  y  cos θ. In particular, we can say that two vectors vectors x and y in R n are perpen dicular or orthogonal if x · y = 0. This provides much intuition for dealing with vectors in R n . Thus if a = ( a 1 ,...a n ) is a nonzero element of R n , the homogeneous linear equation a 1 x 1 + ··· + a n x n = 0 describes the set of all vectors x = ( x 1 ,...,x n ) ∈ R n that are perpendicular to a . The set of solutions W = { x ∈ R n : a · x = 0 } to this homogeneous linear equation is a linear subspace of R n . We remind you that to see that W is a linear subspace, we need to check three facts: 1. a · = 0, so ∈ W . 2. If x ∈ W and y ∈ W , then a · x = 0 and a · y = 0, and it follows from the axioms for dot product that a · ( x + y ) = 0 so x + y ∈ W ....
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This note was uploaded on 06/05/2010 for the course MATH Math 108a taught by Professor Moore during the Spring '10 term at UCSB.
 Spring '10
 MOORE
 Calculus, Linear Algebra, Algebra, Dot Product

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