Linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace[1][note 1] is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
Definition[edit]
If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K. Equivalently, a nonempty subset W is a linear subspace of V if, whenever w1, w2 are elements of W and α, β are elements of K, it follows that αw1 + βw2 is in W.[2][3][4][5][6]
As a corollary, all vector spaces are equipped with at least two (possibly different) linear subspaces: the zero vector space consisting of the zero vector alone and the entire vector space itself. These are called the trivial subspaces of the vector space.[7]
Examples[edit]
Example I[edit]
In the vector space V = R3 (the real coordinate space over the field R of real numbers), take W to be the set of all vectors in V whose last component is 0.
Then W is a subspace of V.
Proof:
Properties of subspaces[edit]
From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples.[8] Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a subspace if and only if every linear combination of finitely many elements of W also belongs to W.
The equivalent definition states that it is also equivalent to consider linear combinations of two elements at a time.
In a topological vector space X, a subspace W need not be topologically closed, but a finite-dimensional subspace is always closed.[9] The same is true for subspaces of finite codimension (i.e., subspaces determined by a finite number of continuous linear functionals).
Operations and relations on subspaces[edit]
Inclusion[edit]
The set-theoretical inclusion binary relation specifies a partial order on the set of all subspaces (of any dimension).
A subspace cannot lie in any subspace of lesser dimension. If dim U = k, a finite number, and U ⊂ W, then dim W = k if and only if U = W.