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Unformatted text preview: Chapter 5: Orthogonality April 30, 2009 Week 1314 1 Inner product Geometric concepts of length, distance, angle, and orthogonality, which are wellknown in R 2 and R 3 , can be defined in R n . These concepts provide powerful geometric tools to solve many applied problems such as the leastsquares problem. Given two straight lines ‘ 1 : y = a 1 x and ‘ 2 : y = a 2 x . We know that ‘ 1 and ‘ 2 are perpendicular, written ‘ 1 ⊥ ‘ 2 , if and only if a 1 a 2 = 1. Note that the directions of the straight lines ‘ 1 and ‘ 2 can be described respectively by the nonzero vectors u = • 1 a 1 ‚ = • u 1 u 2 ‚ and v = • 1 a 2 ‚ = • v 1 v 2 ‚ . Then a 1 a 2 = 1 is equivalent to 1 + a 1 a 2 = 0. Let u · v := u 1 v 1 + u 2 v 2 . Thus ‘ 1 ⊥ ‘ 2 ⇐⇒ u · v = 0 . The inner product (or dot product ) of two vectors u and v in R n is the number h u , v i := u · v = u 1 v 1 + ··· + u n v n , where u = u 1 . . . u n , v = v 1 . . . v n . The transpose of u is the row vector u T = [ u 1 ,...,u n ]. The matrix product u T v is a 1 × 1 matrix, and u T v = [ u · v ] . If we identify any 1 × 1 matrix [ c ] to its entry c , the inner product can be written as the matrix multiplication u · v = u T v . Proposition 1.1. For vectors u , v , w in R n and scalar c , (1) u · v = v · u , (2) ( u + v ) · w = u · w + v · w , (3) ( c u ) · v = c ( u · v ) , 1 (4) u · u ≥ and u · u = 0 if and only if u = . The length (or norm ) of a vector v in R n is the nonnegative number k v k = √ v · v = q v 2 1 + v 2 2 + ··· + v 2 n . It is clear that for any vector v in R n and any scalar c , k c v k =  c k v k . The distance d ( u , v ) between two vectors u and v in R n is the length of the vector u v , i.e., d ( u , v ) = k u v k . Theorem 1.2. Two vectors u and v in R n are orthogonal if and only if k u + v k 2 = k u k 2 + k v k 2 . (1.1) Proof. By linearity of inner product, we have k u + v k 2 = ( u + v ) · ( u + v ) = u · u + v · v + 2 u · v = k u k 2 + k v k 2 + 2 u · v . It is clear that (1.1) is valid if and only if u · v = 0. The vectors u and v are called orthogonal if u · v = 0 . Lemma 1.3. For nonzero vectors u , v of R n , 1 ≤ u · v k u kk v k ≤ 1 . Proof. Consider the vectors w ( t ) = u + t v , where t is a real variable. Then y ( t ) := ( u + t v ) · ( u + t v ) = u · u + 2 t u · v + t 2 v · v ≥ . Thus quadratic function y = y ( t ) is above the taxis, and the equation u · u ) + 2 t u · v + t 2 v · v = 0 has at mots one root. Therefore the discriminant Δ := b 2 4 ac = 4( u · v ) 2 4( u · u )( v · v ) ≤ . This inequality is equivalent to  u · v  ≤ k u kk v k ....
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This note was uploaded on 01/28/2011 for the course MATH 113 taught by Professor Beifangchan during the Fall '08 term at HKUST.
 Fall '08
 BeifangChan
 Calculus

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