V ECTOR S PACES Throughout mathematics we come across many types objects which

V ector s paces throughout mathematics we come across

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3. VECTORSPACESThroughout mathematics we come across many typesobjects which can be added and multiplied by scalars toarrive at similar types of objects.We have observed these forR,C,Mm,n(R)andMm,n(C).Inthe last semester, you have come across with the space ofcontinuous functions, or the space of differentiablefunctions etc.What is the common property to all of these spaces?
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3. VECTORSPACESThroughout mathematics we come across many typesobjects which can be added and multiplied by scalars toarrive at similar types of objects.We have observed these forR,C,Mm,n(R)andMm,n(C).Inthe last semester, you have come across with the space ofcontinuous functions, or the space of differentiablefunctions etc.What is the common property to all of these spaces?
Image of page 29
3. VECTORSPACESThroughout mathematics we come across many typesobjects which can be added and multiplied by scalars toarrive at similar types of objects.We have observed these forR,C,Mm,n(R)andMm,n(C).Inthe last semester, you have come across with the space ofcontinuous functions, or the space of differentiablefunctions etc.What is the common property to all of these spaces? 15/26
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Definition A nonempty set V of objects (called elements or vectors ) is called a vector space over K if the following axioms are satisfied : 16/26
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DefinitionA nonempty set V of objects (calledelementsorvectors) iscalled avector spaceoverKif the following axioms aresatisfied :I.Closure axioms:16/26
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DefinitionA nonempty set V of objects (calledelementsorvectors) iscalled avector spaceoverKif the following axioms aresatisfied :I.Closure axioms:1.(closure under addition) For every x,yV there is aunique x+yV.16/26
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DefinitionA nonempty set V of objects (calledelementsorvectors) iscalled avector spaceoverKif the following axioms aresatisfied :I.Closure axioms:1.(closure under addition) For every x,yV there is aunique x+yV.2.(closure under multiplication by reals) For every xV andscalarαKthere is a unique elementαxV.16/26
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