\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Real Vectors Span
Needed by:
Invariant Real Subspaces
Orthogonal Real Subspaces
Real Affine Sets and Subspaces
Real Convex Sets
Real Matrix Range
Real Matrix Rank
Real Subspace Dimensions
Vector Subspaces
Sheet PDF
Graph PDF

Real Subspaces


A nonempty set $S \subset \R ^n$ is called a subspace (or linear subspace, vector subspace) if

  1. $x + y \in S$ for all $x, y \in S$, and
  2. $\alpha x \in S$ for all $\alpha \in \R $, $x \in S$.
We say that that $S$ is (1) closed under vector addition and (2) closed under scalar multiplication.


The set $S_1 = \R ^n$ is a subspace. In other words, the entire set is a subspace of itself. The set $S_2 = \set{0}$, consisting of a single point, the origin, is a subspace. $S_1$ is the biggest subspace. In other words, if $S'$ is another subspace of $\R ^n$, then $S' \subset S_1$. If $S$ is a subspace, it is nonempty, so there is $x \in S$, and it is closed under scalar multiplication, so $0\cdot x = 0 \in S$. In other words, every subspace contains the origin. So $S_2$ is the smallest subspace, in the sense that if $S'$ is another subspace $S_2 \subset S'$.

The span (see Real Vectors Span) of a set of vectors $v_1, \dots , v_k$ is a subspace. For two subspaces $S, T \subset \R ^n$, their sum

\[ S + T = \Set{x + y}{x \in S, y \in T} \]

is a subspace.

Geometric intuition

Roughly speaking, a subspace $S$ is a flat set which passes through the origin. In $\R ^2$, the subspaces are the lines. In $\R ^3$, the lines and the planes.

Copyright © 2023 The Bourbaki Authors — All rights reserved — Version 13a6779cc About Show the old page view