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la09-rips - LINEAR ALGEBRA W W L CHEN c W W L Chen 1997...

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LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2005. This chapter is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 9 REAL INNER PRODUCT SPACES 9.1. Euclidean Inner Products In this section, we consider vectors of the form u = ( u 1 , . . . , u n ) in the euclidean space R n . In particular, we shall generalize the concept of dot product, norm and distance, first developed for R 2 and R 3 in Chapter 4. Definition. Suppose that u = ( u 1 , . . . , u n ) and v = ( v 1 , . . . , v n ) are vectors in R n . The euclidean dot product of u and v is defined by u · v = u 1 v 1 + . . . + u n v n , the euclidean norm of u is defined by u = ( u · u ) 1 / 2 = ( u 2 1 + . . . + u 2 n ) 1 / 2 , and the euclidean distance between u and v is defined by d ( u , v ) = u v = (( u 1 v 1 ) 2 + . . . + ( u n v n ) 2 ) 1 / 2 . PROPOSITION 9A. Suppose that u , v , w R n and c R . Then (a) u · v = v · u ; (b) u · ( v + w ) = ( u · v ) + ( u · w ) ; (c) c ( u · v ) = ( c u ) · v ; and (d) u · u 0 , and u · u = 0 if and only if u = 0 . Chapter 9 : Real Inner Product Spaces page 1 of 15
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Linear Algebra c W W L Chen, 1997, 2005 PROPOSITION 9B. (CAUCHY-SCHWARZ INEQUALITY) Suppose that u , v R n . Then | u · v | ≤ u v . In other words, | u 1 v 1 + . . . + u n v n | ≤ ( u 2 1 + . . . + u 2 n ) 1 / 2 ( v 2 1 + . . . + v 2 n ) 1 / 2 . PROPOSITION 9C. Suppose that u , v R n and c R . Then (a) u 0 ; (b) u = 0 if and only if u = 0 ; (c) c u = | c | u ; and (d) u + v u + v . PROPOSITION 9D. Suppose that u , v , w R n . Then (a) d ( u , v ) 0 ; (b) d ( u , v ) = 0 if and only if u = v ; (c) d ( u , v ) = d ( v , u ) ; and (d) d ( u , v ) d ( u , w ) + d ( w , v ) . Remark. Parts (d) of Propositions 9C and 9D are known as the Triangle inequality. In R 2 and R 3 , we say that two non-zero vectors are perpendicular if their dot product is zero. We now generalize this idea to vectors in R n . Definition. Two vectors u , v R n are said to be orthogonal if u · v = 0. Example 9.1.1. Suppose that u , v R n are orthogonal. Then u + v 2 = ( u + v ) · ( u + v ) = u · u + 2 u · v + v · v = u 2 + v 2 . This is an extension of Pythagoras’s theorem. Remarks. (1) Suppose that we write u , v R n as column matrices. Then u · v = v t u , where we use matrix multiplication on the right hand side. (2) Matrix multiplication can be described in terms of dot product. Suppose that A is an m × n matrix and B is an n × p matrix. If we let r 1 , . . . , r m denote the vectors formed from the rows of A , and let c 1 , . . . , c p denote the vectors formed from the columns of B , then AB = r 1 · c 1 . . . r 1 · c p . . . . . . r m · c 1 . . . r m · c p . 9.2. Real Inner Products The purpose of this section and the next is to extend our discussion to define inner products in real vector spaces. We begin by giving a reminder of the basics of real vector spaces or vector spaces over R .
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