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==Definition==
Let {{M|S}} be a [[set]] and let {{M|\mathbb{F} }} be a [[field]]. The "''free [[vector space]] over {{M|\mathbb{F} }} generated by {{M|S}}''", denoted {{M|\mathcal{F}_\mathbb{F}(S)}}<ref group="Note" name="notation">From [[Books:Introduction to Smooth Manifolds - John M. Lee]]'s notation {{M|\mathcal{F}(S)}} for the free vec space over {{M|\mathbb{R} }}, we don't specify the field though, so both of these are sensible notations</ref> or {{M|\mathcal{F}(S;\mathbb{F})}}<ref group="Note" name="notation"/> is defined as follows{{rITSMJML}}:{{M|\newcommand{\freevec}[2][\mathbb{F}]{\mathcal{F}(#2;#1)} }}
* {{M|\freevec{S} }} is the set of all [[formal linear combinations]] of elements of {{M|S}}
* {{M|\freevec{S} }} is a [[vector space]] under the operations of [[pointwise]] addition and the obvious scalar multiplication as follows:
*# '''Addition: ''' let {{M|f,g\in\freevec{S} }} then {{M|(f+g):S\rightarrow\mathbb{F} }} is given by {{M|(f+g):s\mapsto f(s)+g(s)}}
*#* This addition is defined as both {{M|f(s),g(s)\in\mathbb{F} }} and a field has addition
*# '''Scalar multiplication: ''' let {{M|\lambda\in\mathbb{F} }} and {{M|f\in\freevec{S} }} be given, then we define {{M|(\lambda f):S\rightarrow\mathbb{F} }} by {{M|(\lambda f):s\mapsto \lambda f(s)}}
*#* Again, {{M|\lambda,f(s)\in\mathbb{F} }} and as {{M|\mathbb{F} }} is a field, this notion of multiplication is defined.
==={{plural|Formal linear combination|s}}===
{{:Formal linear combination/Definition}}
==[[Characteristic property of the free vector space|Characteristic property]]==
{{:Characteristic property of the free vector space/Statement}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Abstract Algebra|Linear Algebra}}