u5 - Unit V: Orthogonality 1. Dot Product and Length...

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Unformatted text preview: Unit V: Orthogonality 1. Dot Product and Length Definition: The dot product of two vectors u , v R n is u v = n X j =1 u j v j . The dot product is symmetric: u v = v u , u , v R n . The dot product depends linearly on u and on v : ( a x + b y ) v = a ( x v ) + b ( y v ) , x , y , v R n , a,b R , u ( a x + b y ) = a ( u x ) + b ( u y ) , x , y , u R n , a,b R . Definition: The length of a vector u R n is defined by || u || = u u = n X j =1 u 2 j ! 1 / 2 , where we take the nonegative square root. Thus || u || 2 = u u = n X j =1 u 2 j . Lemma. The length of a vector in R n has the following properties: (1) || u || 0, (2) || u || = 0 if and only if u = , (3) || c u || = | c ||| u || for u R n and c scalar. Definition: A unit vector is a vector u that has unit length, || u || = 1. If v 6 = , then u = v / || v || is a unit vector. Theorem 1 (Cauchy-Schwarz Inequality). If u , v R n , then | u v | || u |||| v || . Equality holds if and only if one of the vectors is a multiple of the other. Proof. We may assume that v 6 = . We start with the inequality || u + t v || 2 = ( u + t v ) ( u + t v ) , and we expand the right-hand side, to obtain || u || 2 + 2 t u v + t 2 || v || 2 . 1 For fixed u and v 6 = , this is a quadratic polynomial in t . It attains its minimum at t =- u v || v || 2 . We plug this value of t into the inequality, and we obtain || u || 2- 2 ( u v ) 2 || v || 2 + ( u v ) 2 || v || 4 || v || 2 = || u || 2- ( u v ) 2 || v || 2 . Thus ( u v ) 2 || u || 2 || v || 2 , and this is equivalent to the Cauchy-Schwarz inequality. Equality holds in the Cauchy-Schwarz inequality if and only if the minimum of the function || u + t v || 2 is 0. This occurs if and only if u + t v = for some t , and this occurs if and only if u is a multiple of v . Suppose that u and v are nonzero vectors in R n . The Cauchy-Schwarz inequality allows us to define the angle between u and v to be the value of between 0 and such that cos = u v || u |||| v || . Then u v = || u |||| v || cos . Note that = 0 if and only if both u v > 0 and equality holds in the Cauchy-Schwarz inequality. This occurs if and only if u and v are positive multiples of each other. Similarly, = if and only if u and v are negative multiples of each other. Theorem 2 (Triangle Inequality). For any two vectors u , v R n , || u + v || || u || + || v || . Proof. Use || u + v || 2 = ( u + v ) ( u + v ) = || u || 2 + 2 u v + || v || 2 and the Cauchy-Schwarz inequality, to obtain || u + v || 2 || u || 2 + 2 || u |||| v || + || v || 2 = ( || u || + || v || ) 2 ....
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u5 - Unit V: Orthogonality 1. Dot Product and Length...

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