36 - 16.1-16.2January 17, 20121The dot productDefinition...

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Unformatted text preview: 16.1-16.2January 17, 20121The dot productDefinition 1.1.Avector spaceis a setVwith two mappingsVVVandRVVcalled vector addition and scalar multiplication such that for alla,bVand,Ri)a+ (b+c) = (a+b) +cii)a+b=b+aiii)a+=aiv)a+ (-a) =v)()a=(a)vi)(+)a=a+bvii)(a+b) =a+bviii)1a=aRemark 1.1.What about notions of multiplication for vectors?Definition 1.2.Aninner producton a vector spaceVis a functionVVRwhich associates with each pair(a,b)of vectors inVa real number<a,b>whichsatisfies1)<a,a>>ifa6= 0(positivity)2)<a,b>=<b,a>(symmetry)3)< a+b,c>= <a,c>+ <b,c>1Definition 1.3.Thedot productwhich is the usual inner product on the vectorspaceRnis defined byab=ni=1aibi.Remark 1.2.The dot product yields a notion of length or size of a vector|a|=(aa)1/2, inRnthis notion is theEuclidean norm.Definition 1.4.TheEuclidean distancebetween two vectorsa,bRnisdefined to be|a-b|....
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This note was uploaded on 01/18/2012 for the course MATH 255 taught by Professor Jackwaddell during the Winter '08 term at University of Michigan.

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36 - 16.1-16.2January 17, 20121The dot productDefinition...

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