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?

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?

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

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

DefinitionA nonempty set V of objects (calledelementsorvectors) iscalled avector spaceoverKif the following axioms aresatisfied :I.Closure axioms:16/26

DefinitionA nonempty set V of objects (calledelementsorvectors) iscalled avector spaceoverKif the following axioms aresatisfied :I.Closure axioms:1.(closure under addition) For every x,y∈V there is aunique x+y∈V.16/26

DefinitionA nonempty set V of objects (calledelementsorvectors) iscalled avector spaceoverKif the following axioms aresatisfied :I.Closure axioms:1.(closure under addition) For every x,y∈V there is aunique x+y∈V.2.(closure under multiplication by reals) For every x∈V andscalarα∈Kthere is a unique elementαx∈V.16/26

#### You've reached the end of your free preview.

Want to read all 97 pages?

- Spring '16
- Ravi Banavar
- Linear Algebra, Algebra, Determinant, Scalar, Vector Space, linear span