Transcription of Vectors and Vector Spaces
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Vectors and Vector Spaces
Chapter 1 Vectors and Vector Vector SpacesUnderlying every Vector space (to be defined shortly) is a of scalarfields are the real and the complex numbersR:= real numbersC:= complex are the onlyfields we use spaceVis a collection of objects with a ( Vector )addition and scalar multiplication defined that closed under both operationsand which in addition satisfies the following axioms:(i) ( + )x= x+ xfor allx Vand , F(ii) ( x)=( )x(iii)x+y=y+xfor allx, y V(iv)x+(y+z)=(x+y)+zfor allx, y, z V(v) (x+y)= x+ y(vi) O Vz0+x=x; 0 is usually called theorigin(vii) 0x=0(viii)ex=xwhereeis the multiplicative unit 1. Vectors AND Vector SPACESThe closed property mentioned above means that for all , Fandx, y V x+ y V( you can t leaveVusing Vector addition and scalar multiplication). Also,when we write for , Fandx V( + )xthe + is in thefield, whereas when we writex+yforx, y V,the + isin the Vector space.
Vectors and Vector Spaces 1.1 Vector Spaces Underlying every vector space (to be defined shortly) is a scalar field F. ... (viii) ex = x where e is the multiplicative unit in F. 7. 8 CHAPTER 1. VECTORS AND VECTOR SPACES The “closed” property mentioned above means that for all α,β∈F and x,y ∈V
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