Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space. A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of .

Let K be a field (such as the real numbers), and let V be a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. Ignoring the full extent of mathematical generalization, scalars can be understood simply as numbers. Suppose that W is a subset of V. If W is a vector space itself (which means that it is closed under operations of addition and scalar multiplication), with the same vector space operations as V has, then W is a subspace of V.

To use this definition, we don't have to prove that all the properties of a vector space hold for W. Instead, we can prove a theorem that gives us an easier way to show that a subset of a vector space is a subspace.

Theorem: Let V be a vector space over the field K, and let W be a subset of V. Then W is a subspace if and only if W satisfies the following three conditions:

Proof: Firstly, property 1 ensures W is nonempty. Looking at the definition of a vector space, we see that properties 2 and 3 above assure closure of W under addition and scalar multiplication, so the vector space operations are well defined. Since elements of W are necessarily elements of V, axioms 1, 2 and 5–8 of a vector space are satisfied. By the closure of W under scalar multiplication (specifically by 0 and −1), the vector space's definitional axiom identity element of addition and axiom inverse element of addition are satisfied.

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