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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 Spaces 1.1 Vector Spaces Underlying every vector space (to be defined shortly) is a scalar field F. ... If we allow all the scalars to be zero we can always arrange for (T) to hold, making the concept vacuous. Proposition 1.2.1. If …

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