Linear subspace

A linear subspace is a subset of a vector space that contains the zero vector and is closed under vector addition and scalar multiplication. With these inherited operations, it is itself a vector space over the same field as the containing space. Subspaces are fundamental objects in linear algebra: they describe directions, solution sets of homogeneous equations, and smaller spaces embedded within larger ones. Their elements need not be geometric arrows; they may also be matrices, polynomials, or functions. (math.brown.edu)

Definition and subspace test

Let $V$ be a vector space over a field $F$. A subset $W\subseteq V$ is a subspace if:

1. $0\in W$;
2. $u+v\in W$ whenever $u,v\in W$;
3. $av\in W$ whenever $a\in F$ and $v\in W$.

Equivalently, a nonempty subset is a subspace precisely when

This criterion says that $W$ is closed under linear combinations. Nonemptiness matters: the empty set satisfies the closure conditions vacuously but is not a vector space. Additive inverses follow by choosing the scalar $-1$, while the remaining vector-space axioms are inherited from $V$. (math.brown.edu)

The scalar field is part of the definition. For example, the real numbers form a subspace of the complex numbers when the latter are regarded as a real vector space, but not when they are regarded as a complex vector space: multiplication by $i$ does not preserve the real numbers. (homepages.ucl.ac.uk)

Geometric and algebraic examples

Every vector space has the zero subspace $\{0\}$ and the whole space $V$. A proper subspace is one unequal to $V$. In $\mathbb R^2$, the subspaces are the origin, lines through the origin, and the entire plane. In $\mathbb R^3$, planes through the origin are additional possibilities. In particular,

is a plane subspace, since addition and scalar multiplication preserve its defining equation. (ocw.mit.edu)

A line or plane not containing the origin is not a linear subspace. A translated set $v_0+W=\{v_0+w:w\in W\}$ is instead an affine subspace; it is linear exactly when $v_0\in W$. Thus $x+y+z=1$ defines an affine plane rather than a linear one. (math.brown.edu)

Subspaces also arise without geometric coordinates. The polynomials of degree at most $d$, together with the zero polynomial, form a subspace of $F[x]$. Likewise, symmetric real matrices form a subspace of the space of square matrices of a fixed size. These examples illustrate that the defining issue is closure under operations, not the appearance of the elements. (math.brown.edu)

Span, basis, and dimension

For a collection of vectors $S\subseteq V$, its linear span consists of all finite linear combinations of elements of $S$. It is the smallest subspace containing $S$. Consequently, specifying spanning vectors is one way to describe a subspace. (homepages.ucl.ac.uk)

A basis of $W$ is a spanning collection satisfying linear independence. Its size defines the dimension of $W$. If $V$ is finite-dimensional, then

Equality $\dim W=\dim V$ forces $W=V$. Every basis of $W$ can be extended to a basis of $V$, distinguishing directions already present in the subspace from additional directions in the ambient space. (homepages.ucl.ac.uk)

For the plane $x+y+z=0$, one possible basis is

Every point in the plane is a unique linear combination of these two vectors, so the plane has dimension two. (ocw.mit.edu)

Intersections, sums, and complements

The intersection of any nonempty family of subspaces of $V$ is again a subspace. The union of two subspaces generally is not: adding a vector from one to a vector from the other may leave the union. For two subspaces $U,W$, their union is a subspace exactly when one is contained in the other. (cfm.brown.edu)

Their sum is

the smallest subspace containing both. For finite-dimensional subspaces,

The subtraction accounts for directions shared by the two spaces. (cfm.brown.edu)

If $U\cap W=\{0\}$, each element of the sum has a unique decomposition $u+w$; the sum is then called a direct sum, written $U\oplus W$. A complement of $W$ in $V$ is a subspace $U$ for which $V=W\oplus U$. Every subspace of a finite-dimensional vector space has a complement, though it is generally not unique. (cfm.brown.edu)

Linear maps and equations

For a linear map $T:V\to Y$, its kernel $\ker T=\{v:T(v)=0\}$ is a subspace of $V$, and its image is a subspace of $Y$. For a matrix $A$, these become its nullspace and column space. A homogeneous system of linear equations $Ax=0$ therefore always has a subspace as its solution set. (ocw.mit.edu)

The rank–nullity theorem relates their dimensions:

when $V$ is finite-dimensional. For an $m\times n$ matrix, the nullspace consequently has dimension $n-\operatorname{rank}(A)$, where rank is the dimension of the column space. If $Ax=b$ has a particular solution $x_0$, its complete solution set is $x_0+\ker A$, generally an affine rather than linear subspace. (ocw.mit.edu)

Orthogonality and approximation

An inner product provides a distinguished complement in finite dimensions. The orthogonal complement

is a subspace, and $V=W\oplus W^\perp$. Each vector decomposes uniquely into a component in $W$ and a perpendicular component. (cfm.brown.edu)

The first component is its orthogonal projection onto $W$, the unique nearest point of $W$ in the induced norm. In ordinary least squares, fitted vectors are projections of the observed response vector onto the design matrix’s column space; residuals lie in its orthogonal complement. This connects subspace geometry directly with approximation and regression. (ocw.mit.edu)