Vector subspace

A vector subspace is a nonempty subset of a vector space closed under vector addition and scalar multiplication, and is therefore itself a vector space. Equivalently, whenever $u,v$ belong to the subset, so does $au+bv$ for all allowed scalars $a,b$. In particular it contains the zero vector.

For example, the plane $W=\{(x,y,z):x+y+z=0\}\subseteq {\mathbb{R}}^3$ is a vector subspace, with basis $(1,-1,0),(1,0,-1)$. Indeed every element is $(-y)(1,-1,0)+(-z)(1,0,-1)$. The parallel plane $x+y+z=1$ misses the zero vector and is an affine subspace instead. Homogeneous constraints produce vector subspaces: $W$ is the kernel of the linear map $(x,y,z)\mapsto x+y+z$. This lets us describe all solutions of a homogeneous linear equation through a basis of its kernel.

For a chosen basis $(e_1,\dots ,e_n)$, the vector subspace spanned by a specified collection of basis vectors. In the standard basis of Euclidean space, its vectors have zero coordinates outside $\mathcal{B}$. The block coordinate gradient is the orthogonal projection of the full gradient onto this subspace. The fixed coordinate subspace consisting of the first $k$ coordinates is a particular example.

A proper vector subspace of a vector space is one strictly smaller than the ambient space. In a finite-dimensional vector space, this is equivalent to having smaller dimension: a basis of the subspace can be extended to a basis of the ambient space, and strict inclusion requires at least one additional vector.

A flag is a strictly nested chain of vector subspaces of a fixed vector space. A complete flag contains a subspace of every possible dimension. For an action by linear maps, an invariant flag means that each of its vector subspaces is an invariant subspace. A basis adapted to an invariant complete flag makes all these maps upper triangular, connecting complete flags to simultaneous triangularization of a Lie algebra representation.

A complete flag in an $n$-dimensional vector space is a nested sequence of vector subspaces with $\dim V_i=i$. An action preserves such a flag exactly when its matrices are upper triangular in a basis adapted to the flag. The Lie theorem supplies an invariant complete flag for every complex finite-dimensional Lie algebra representation of a solvable Lie algebra.

The fixed coordinate subspace of dimension $k$ in ${\mathbb{R}}^p$ consists of vectors whose coordinates after the first $k$ vanish. Restricting a quadratic form to it uses only the leading $k$ by $k$ matrix block. It specifies one fixed support; allowing every support of size $k$ would introduce another combinatorial counting step.

The intersection of any family of vector subspaces of a common vector space is the set of vectors belonging to every member of the family. It is again a vector subspace.

The sum of vector subspaces $U,W\le V$ isIt is the smallest vector subspace containing both $U$ and $W$.

For vector subspaces $U\subseteq W\subseteq V$ of a finite-dimensional vector space, such a $W^{\prime }$ always exists. Extend a basis of $U$ to one of $W$, then extend again to one of $V$. Let $W^{\prime }$ be spanned by the original basis of $U$ together with the vectors added in the second extension. Uniqueness of coordinates gives the displayed intersection and sum. Equivalently, choose a direct sum complement $C$ with $V=W\oplus C$ and put $W^{\prime }=U\oplus C$. Necessarily $\dim W^{\prime }=\dim V-\dim W+\dim U$. In $M_2(\mathbb{R})$, the skew-symmetric subspace $U$ inside the traceless subspace $W$ admits $W^{\prime }=\mathrm{span}\{I,\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\}$.

For vector subspaces $U,W$ of a common vector space, this is a surjective linear map. Its kernel of a linear map is $\{(x,-x):x\in U\cap W\}$, which has a linear isomorphism to $U\cap W$ by $x\mapsto (x,-x)$. The rank-nullity theorem and the direct sum basis give the dimension formula for a sum of subspaces. The paired domain keeps the two contributions separate even when their underlying vectors overlap.

A hyperplane is a vector subspace of codimension one. A projective hyperplane in ${\mathbb{P}}^n$ is the zero set of one nonzero homogeneous linear form.

A closed vector subspace of a topological vector space is both a vector subspace and a closed set. If a continuous linear map has a Hausdorff target, its kernel of a linear map is a closed vector subspace, since it is the preimage of the closed singleton $\{0\}$. The target condition matters because our topological vector space convention permits non-Hausdorff spaces.

For a counterexample without the target hypothesis, the identity linear map of $\mathbb{R}$ with the indiscrete topology is continuous, but its kernel of a linear map $\{0\}$ is not closed.

If $E$ is a nonzero closed linear subspace of a normed ambient space and $F$ is finite-dimensional with $E\cap F=\{0\}$, the unit sphere of $E$ has positive distance from $F$. Otherwise a sequence of nearly coincident points yields a bounded sequence in $F$, hence a convergent subsequence; closedness of $E$ puts its norm-one limit in both vector subspaces. In applications to a bidual space, completeness of the Banach space makes its canonical image closed.