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.

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 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.

Dimension


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