Vector space

Idea
A vector space generalizes the idea of a linear relationship. Normally with real numbers we have the linearity through the distributive property:

$$ a \times (b + c) = (a \times b) + (a \times c) $$

Multiplication is a form of scaling which acts evenly across a number. We would like to expand this notion of linear scaling to objects other than just real numbers. For example there is some sense in which we can scale polygons up and down.

A vector space is one such generalization which allows us to have a set of scalars acting on a set of vectors, with some constraints to ensure that it behaves like we expect a scaling operation to behave.

Most commonly our scalars are real numbers although other types of scalars can exist. The definition allows our set of scalars to be any field, such as $$\mathbb{C}$$. A vector space with real scalars can be called a $$\mathbb{R}$$-vector space.

Classic definition
A vector space is classically defined to be an abelian group $$V = (V,+_V,0_V,-_F)$$, these are our vectors, and field $$F=(F,+_F,0_F,-_F,\times_F,1_F,/_F)$$, our scalars, along with an operation $$* : F \times V \rightarrow V$$ such that:


 * $$(f \times_F g) * v = f * (g * v)$$
 * $$f * (v +_V w) = (f * v) +_V (f * w)$$
 * $$(f +_F g) * v = (f * v) +_V (g * v)$$
 * $$1_F * v = v$$

Module definition
Since the classic definition does not utilize the $$/_F$$ operator it is common to generalize the concept of vector space to a module, which obeys the same laws but requires $F$ only to be a ring. Thus another definition for vector space can be a $R$-module where $R$ is a field.

Homomorphism definition
A vector space can also be defined as ring homomorphism $$ from a field $F$ to the endomorphism ring of an abelian group $V$.

$$ $$
 * : F \rightarrow \mathrm{End}(V)

While this definition is conceptually complex it is short and helps to show why the concept of a vector space is more than just an arbitrary collection of axioms.

A vector space is a way to get a field to act like endomorphisms on an abelian group.

Example
If we take the abelian group $$\mathbb{R}\times\mathbb{R}$$, where $$\mathbb{R}$$ is the real numbers under addition, then its endomorphism ring is on the set of functions $$M : \mathbb{R}\rightarrow \mathbb{R}$$ such that $$f(x+y) = f(x) + f(y)$$. That is the set linear functions on pairs of real numbers. In other words it is precisely the set of $$2\times2$$ matrices of real numbers. The addition and multiplication are matrix addition and function composition, meaning that $$\mathrm{End}(\mathbb{R}\times\mathbb{R})$$ is isomorphic to the usual ring of matrices.

From there we can select the following homomorphism:

$$ $$
 * : \mathbb{R} \rightarrow \mathrm{End}(\mathbb{R}\times\mathbb{R}) \\
 * = a \mapsto \begin{bmatrix}a & 0 \\ 0 & a\end{bmatrix}

In fact this homomorphism is unique and it is thus the only $$\mathbb{R}$$-vector space that can be formed on the abelian group $$\mathbb{R}\times\mathbb{R}$$.

Linear combination
A vector $u$ is a linear combination of some sequence of vectors $$v_0^n$$ iff there exists some sequence of scalars $$a_0^n$$ such that:

$$ u = \sum_{i=0}^na_i*v_i $$

Since vector addition is commutative it is almost always possible to talk about linear combinations of a set instead of a sequence. As a result, the set language is the more common terminology.

Linear independence
A sequence of vectors $$v_0^n$$ is linearly dependent iff there exists some sequence of scalars $$a_0^n$$ such that, at least one of the scalars is non-zero, and:

$$ 0 = \sum_{i=0}^na_i*v_i $$

A sequence of vectors which is not linearly dependent is called linearly independent. As with linear combinations, it is common to talk about the linear independence of a set of vectors. A set of vectors is called linearly independent if no two distinct vectors in the set are linearly dependent.

Span
The span of some set of vectors is the set of all linear combinations of that set.

Basis
A basis of a vector space is a linearly independent set of non-zero vectors such that its span is the vector space.

The basis of a vector space is very rarely unique, however all bases of a vector space have the same size.

The statement "Every vector space has at least one basis." is equivalent to the axiom of choice. Thus if you assume the axiom of choice every vector space has at least one basis and its dimension is defined. An example of a vector space which requires the axiom of choice to create a basis can be found in the examples.

Dimension


The dimension of a vector space is the size of any basis set.

Convexity
A notion of convexity can be defined for a $$\mathbb{R}$$-vector space.

A convex set is a set of vectors $S$ such that for any two vectors $$v_0 \in S$$, $$v_1 \in S$$, the set:

$$ \left\{a*(v_1-v_0)+v_0\mid a\in[0,1]\right\} $$

is a subset of $S$. Informally this is to say that for any two vectors in $S$ the line between them is a subset of $S$.

This notion of convexity is useful for the study of convex polytopes, however it is not the only notion of convexity and authors may use incompatible or generalized notions of convexity.

Direct product
The direct product is a way to form a new $F$-vector space out of two existing $F$-vector spaces. The direct product of two $F$-vector spaces $V$ and $W$ is a $F$-vector space with its vector group being the direct product of the vector groups of $V$ and $W$ and its scalar multiplication being defined as:

$$ a*_{V\times W}(v,w) = (a*_Vv,a*_Ww) $$

Using this direct product the vector spaces $$\mathbb{R}^n$$ can be defined. First $$\mathbb{R}^1$$ is defined to be a $$\mathbb{R}$$-vector space acting on the additive group of $$\mathbb{R}$$. With the scalar product being just multiplication. Then

$$ \mathbb{R}^{n+1} = \mathbb{R}^{n}\times \mathbb{R}^1 $$

Trivial vector space
The simplest possible vector space is the trivial vector space. For any field $F$ there is a $F$-vector space, whose only element the zero vector. This vector space has dimension 0. It may be called $$F^0$$ for some field $F$. For example $$\mathbb{R}^0$$ can represent the zero dimensional real vector space.

Simple vector spaces
For any field $F$ there is also a vector space whose vectors are simply elements of $F$ with the scalar product $$ being the multiplication operation from the field. The direct product operation then ensures a family of vector spaces $$F^n$$ made by combining $n$ copies of this vector space with the direct product.

Concretely the vector spaces $$\mathbb{R}^n$$ and $$\mathbb{C}^n$$ are common, especially in the study of polytopes.

The complex numbers
Even though the complex numbers are a field themselves they are also are an example of a $$\mathbb{R}$$-vector space. The scalar operation is simply the ordinary multiplication operation for complex numbers. The set $$\{1,i\}$$ forms a basis for this vector space since any complex number can be written in the form $$a+bi$$. Thus the dimension of this space is $2$, corresponding to the idea that you can draw the complex numbers as a plane.

More generally if $K$ is a subfield of $F$ then there is a $K$-vector space whose vectors are those elements of $F$, with the scalar multiplication being the multiplicative operation of $F$.

The real numbers
As a result of the last fact the real numbers are an example of a $$\mathbb{Q}$$-vector space. However unlike the last example there is no finite basis for this vector space. Forming a basis at all requires the axiom of choice, and yields a basis of size $$\beth_1$$, the same size as the real numbers themselves.

Function spaces
It's common to represent vectors in certain vector spaces as tuples. For example vectors in $$\mathbb{R}^2$$ can be represented as $$(x,y)$$ pairs. Another way to conceptualize them is as functions from an index to a value. For example vectors in $$\mathbb{R}^2$$ can be functions $$v : \{0,1\} \rightarrow \mathbb{R}$$, with $$v(0)=x$$ and $$v(1)=y$$. This fact extends more generally. The set of functions $$X \rightarrow F$$ where $F$ is a field can form a $F$-vector space where

$$ v + w = x \mapsto v(x) +_F w(x) \\ f * v = x \mapsto f \times_F v(x) $$

This allows not only an alternative method of defining $$\mathbb{R}^n$$, as functions $$\{1,\dots,n\}$$, but it also allows us to define some infinite dimensional spaces such as the $$\mathbb{R}$$-vector space of functions $$\mathbb{Z}\rightarrow\mathbb{R}$$, often called $$\mathbb{R}^\infty$$. The particular space of functions $$\mathbb{R}\rightarrow\mathbb{R}$$ is an important vector space in analysis.