# Vector Space vs. Vector Field A [vector space](Abstract%20Vector%20Spaces.md) can be briefly defined as: > A **vector space** is a *set* of objects called _vectors_, which may be *added* together and *multiplied* ("scaled") by numbers called *scalars* (/posts/where scalars can generally be from any [field](Field.md). A [vector field](Vector-Fields.md) can be briefly defined as: > A type of [field](Field.md) (so, a *set* that has an addition and multiplication operation defined) where each point in this set is associated with a vector. At first these definitions may seem incredibly similar; they both deal with *sets* of elements that have some sort of *addition* and *multiplication* defined. However, it is the operator of *multiplication* that produces their differences #### Field Operations $+ : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}$ $\times : \mathbb{F} \times \mathbb{F} \rightarrow \mathbb{F}$ #### Vector Space Operations (over a field $\mathbb{F}$) $+ : \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{V}$ $\color{green} \times : \mathbb{F} \times \mathbb{V} \rightarrow \mathbb{V}$ So, we can see that for a vector space, the concept of multiplication involves in mapping an element of the field, $\mathbb{F}$ (think scalar), and an element of the vector space, and returning an element of the vector space. #### Consequent Differences... One of the field axioms says that any nonzero element $f \in \mathbb{F}$ has a multiplicative inverse, namely an element $f^{-1} \in \mathbb{F}$ such that $f \times f^{-1} = 1 = f^{-1} \times f$. There is no corresponding property among the vector space axioms. It's an important example---and possibly the source of the confusion between these objects---that any field $\mathbb{F}$ is a vector space over itself, and in this special case the operations $\cdot$ and $\times$ coincide (see more [here](https://math.stackexchange.com/questions/493588/prove-that-the-field-f-is-a-vector-space-over-itself)). #### More intuitive differences A vector space is a set of possible vectors. A vector field is, loosely speaking, a map from some set into a vector space. --- A vector space is something like actual space - a bunch of points. A vector field is an association of a vector with every point in actual space. --- A vector space you draw as a coordinate system. A vector field you draw as a bunch of vectors all over your plane/space/etc. --- Links to: [Abstract Vector Spaces](Abstract%20Vector%20Spaces.md) [Vector-Fields](Vector-Fields.md) [Field](Field.md) References: * [Quora answer](https://www.quora.com/What-is-the-difference-between-a-vector-field-and-a-vector-space) *