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\begin{document}

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\section*{vector space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_linear_algebra}{}\paragraph*{{Higher linear algebra}}\label{higher_linear_algebra}

[[!include homotopy - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{multisorted_notion}{Multisorted notion}\dotfill \pageref*{multisorted_notion} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

For $k$ a [[field]] or a [[division ring]], a \textbf{vector space} over $k$ (or a $k$-vector space) is a [[module]] over the [[ring]] $k$. When the vector space is fixed, its elements are called \emph{vectors}, the field $k$ is referred to as the base field of the ground field of the vector space, and the elements of $k$ are called \emph{scalars}.

Sometimes a vector space over $k$ is called a \textbf{$k$-linear space}. (Compare `$k$-[[linear map]]'.) If $k$ is only a division ring then we carefully distinguish the left $k$-vector spaces and right $k$-vector spaces.

The [[category]] of vector spaces is typically denoted [[Vect]], or $Vect_k$ if we wish to make the field $k$ (the \emph{ground field}) explicit. So

\begin{displaymath}
Vect_k \coloneqq k Mod
  \,.
\end{displaymath}
This category has vector spaces over $k$ as objects, and $k$-linear maps between these as morphisms.

\hypertarget{multisorted_notion}{}\subsubsection*{{Multisorted notion}}\label{multisorted_notion}

Alternatively, one sometimes defines ``vector space'' as a two-sorted notion; taking the field $k$ as one of the sorts and a module over $k$ as the other. More generally, the notion of ``module'' can also be considered as two-sorted, involving a ring and a module over that ring.

This is occasionally convenient; for example, one may define the notion of [[topological vector space]] or topological module as an [[internalization]] in $Top$ of the multisorted notion. This procedure is entirely straightforward for topological modules, as the notion of module can be given by a two-sorted Lawvere theory $T$, whence a topological module (for instance) is just a product-preserving functor $T \to Top$. One may then define a topological vector space as a topological module whose underlying (discretized) ring sort is a field.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

By the \emph{[[basis theorem]]} (and using the [[axiom of choice]]) every vector space admits a [[basis of a vector space|basis]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{vector space}, [[dual vector space]],

\begin{itemize}%
\item [[finite-dimensional vector space]]


\item [[real vector space]], [[complex vector space]]


\item [[topological vector space]], [[convenient vector space]]


\item [[virtual vector space]]



\end{itemize}

\item [[tensor product of vector spaces]]


\item [[real structure]], [[complex structure]], [[quaternionic structure]]


\item [[vector bundle]],


\item [[lattice in a vector space]]


\item [[2-vector space]], [[n-vector space]]


\item [[inner product space]]


\item [[linear operator]], [[matrix]], [[determinant]], [[eigenvalue]], [[eigenvector]]



\end{itemize}
\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

The vector spaces seem to have been first introduced in

\begin{itemize}%
\item Giuseppe Peano, \emph{Calcolo Geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva}, Torino 1888

\end{itemize}
The literature on vector spaces is now extremely large, including lots of elementary linear algebra textbooks. Classics include

\begin{itemize}%
\item [[Michael Artin]], Algebra
\item Israel M. Gelfand, Lectures on linear algebra
\item P. R. Halmos, Finite dimensional vector spaces
\item [[M M Postnikov]], Lectures on geometry, semester 2: Linear algebra

\end{itemize}
Affine spaces are sets which are torsors over the abelian group of vectors of a vector space. Thus vector spaces may serve as a basis for the affine and for the Euclidean geometry. This approach has been invented by [[Hermann Weyl]] in 1918. Dieudonné wrote an influential book on such an approach to 2d and 3d Euclidian geometry, in which the basics of vector spaces in low dimension is introduced along the way (the book is intended for high school teachers):

\begin{itemize}%
\item Jean Alexandre Dieudonn\'e{}, Linear algebra and geometry

\end{itemize}
[[!redirects vector space]] [[!redirects vector spaces]] [[!redirects linear space]] [[!redirects linear spaces]]

[[!redirects Grassmannian]]



\end{document}