This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 16.116.2January 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 vectora=(aa)1/2, inRnthis notion is theEuclidean norm.Definition 1.4.TheEuclidean distancebetween two vectorsa,bRnisdefined to beab....
View
Full
Document
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.
 Winter '08
 JackWaddell
 Addition, Multiplication, Scalar, Vector Space, Dot Product

Click to edit the document details