Re: [Phys-L] fundamental notion of force --> using an arrow to represent something more than a vector

[Diego Saravia <dsa@unsa.edu.ar>]

So you can not define an individual vector

you must define a set, the vector space. all its members are vectors

in that set a sum of two vectors MUST be a vector for example.



wikipedia

A *vector space* (also called a *linear space*) is a collection of objects
called *vectors*, which may be added
<https://en.wikipedia.org/wiki/Vector_addition> together and multiplied
<https://en.wikipedia.org/wiki/Scalar_multiplication> ("scaled") by
numbers, called *scalars
<https://en.wikipedia.org/wiki/Scalar_(mathematics)>* in this context.
Scalars are often taken to be real numbers
<https://en.wikipedia.org/wiki/Real_number>, but there are also vector
spaces with scalar multiplication by complex numbers
<https://en.wikipedia.org/wiki/Complex_number>, rational numbers
<https://en.wikipedia.org/wiki/Rational_number>, or generally any field
<https://en.wikipedia.org/wiki/Field_(mathematics)>. The operations of
vector addition and scalar multiplication must satisfy certain
requirements, called *axioms <https://en.wikipedia.org/wiki/Axiom>*, listed
below <https://en.wikipedia.org/wiki/Vector_space#Definition>.

*Axiom**Meaning*Associativity <https://en.wikipedia.org/wiki/Associativity> of
addition*u* + (*v* + *w*) = (*u* + *v*) + *w*Commutativity
<https://en.wikipedia.org/wiki/Commutativity> of addition*u* + *v* = *v* +
*u*Identity element <https://en.wikipedia.org/wiki/Identity_element> of
additionThere exists an element *0* ∈ *V*, called the *zero vector
<https://en.wikipedia.org/wiki/Zero_vector>*, such that *v* + *0* = *v* for
all *v* ∈ *V*.Inverse elements
<https://en.wikipedia.org/wiki/Inverse_element> of additionFor every *v* ∈ V,
there exists an element −*v* ∈ *V*, called the *additive inverse
<https://en.wikipedia.org/wiki/Additive_inverse>* of *v*, such that *v* + (−
*v*) = *0*.Compatibility <https://en.wikipedia.org/wiki/Semigroup_action> of
scalar multiplication with field multiplication*a*(*b**v*) = (*ab*)*v* [nb
2] <https://en.wikipedia.org/wiki/Vector_space#cite_note-3>Identity element
of scalar multiplication1*v* = *v*, where 1 denotes the multiplicative
identity <https://en.wikipedia.org/wiki/Multiplicative_identity> in *F*.
Distributivity <https://en.wikipedia.org/wiki/Distributivity> of scalar
multiplication with respect to vector addition *a*(*u* + *v*) = *a**u* + *a*
*v*Distributivity of scalar multiplication with respect to field addition(
*a* + *b*)*v* = *a**v* + *b**v*



2015-10-23 23:18 GMT-03:00 Diego Saravia <dsa@unsa.edu.ar>:

> 2015-10-23 19:32 GMT-03:00 John Denker <jsd@av8n.com>:
>
> > On 10/23/2015 03:14 PM, Jeffrey Schnick wrote:
> >
> > > Sometimes a force can be represented by a vector.
>
> a vector is a vector in a specific vector space
>
> different lines of actions are diferent vector spaces in some cases
>
> differents points of applications in other cases
>
> or all the pysical space in others
>
> perhaps these is usefull
>
>  https://en.wikipedia.org/wiki/Affine_space



-- 
Diego Saravia
Diego.Saravia@gmail.com
NO FUNCIONA->dsa@unsa.edu.ar

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