= Vector space
{title2=$V$}
{wiki}
A vector space over a <field> $\mathbb F$ is a <set> $V$ equipped with <vector addition> and <scalar multiplication> satisfying the <vector space axioms>. Its <vectors> can represent directed displacements, lists of measurements, <polynomials>, or entire <functions>. The common structure is that adding two elements and multiplying an element by a <scalar> remain meaningful operations inside the same space.
For example, $\mathbb R^2$ uses coordinatewise addition and scaling. With
$$
b_1=(2,1),\qquad b_2=(1,2),\qquad v=(5,4),
$$
we have $v=2b_1+b_2$. Moving twice along $b_1$ and once along $b_2$ reaches the same <vector> as five horizontal and four vertical steps. The two descriptions refer to different <bases> of the same space. A <basis> makes such descriptions unique; a <spanning set> makes them possible.
\Image[/vector-basis-coordinates.png]
{title=The same vector has coordinates 5,4 in the Cartesian basis and 2,1 in an oblique basis}
The space $\mathbb R[x]_{\leq2}$ of real <polynomials> of degree at most two has <basis> $1,x,x^2$: the <polynomial> $3-2x+x^2$ is the <linear combination> with coordinates $(3,-2,1)$. The <zero vector> here is the zero <polynomial>, and addition combines <coefficients>. A <derivative> is a <linear map> on this space; its <matrix> records how it acts on the three <basis vectors>. The same coordinate machinery therefore handles both geometric motion and operations on <functions>.
The choice of <field> matters. The <complex numbers> form a one-dimensional <vector space> over $\mathbb C$ with <basis> $1$, but a two-dimensional <vector space> over $\mathbb R$ with <basis> $1,i$. The question "how many independent directions?" has an answer only after the allowed <scalars> are specified. Likewise, polynomials of degree exactly two do not form a <vector space>: subtraction can remove the leading term, and the zero <polynomial> is absent.
The usefulness of the abstraction is that a <linear map> is determined by its values on a <basis>. Once those images are known, <linear combinations> give its action on every <vector>. The <rank-nullity theorem> relates the number of attainable outputs to the number of invisible input directions, while <change of basis> lets us choose coordinates suited to the problem. <Eigenvectors>, for example, choose directions on which repeated application of a <linear operator> becomes scalar multiplication.
A historical source is Hermann Grassmann's \i[Die lineale Ausdehnungslehre] (1844). Its full title advertises applications to other branches of mathematics, statics, mechanics, magnetism and crystallography; the original edition is digitized by ETH-Bibliothek. This gives an early setting for treating geometric quantities algebraically across different applications, rather than attributing the modern axiom list to that title alone.https://www.e-rara.ch/zut/content/titleinfo/1169335[Grassmann, Die lineale Ausdehnungslehre, Leipzig, 1844; ETH-Bibliothek Rar 5128]{ref}
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