Subspaces

The objects of interest in geometry are not only the points of the space under consideration, but also its lines, planes, etc. We proceed to study the analogues, in general vector spaces, of these higher-dimensional elements.

Definition 1. A non-empty subset of a vector space is a subspace or a linear manifold if along with every pair, and , of vectors contained in , every linear combination is also contained in .

A word of warning: along with each vector , a subspace also contains . Hence if we interpret subspaces as generalized lines and planes, we must be careful to consider only lines and planes that pass through the origin.

A subspace in a vector space is itself a vector space; the reader can easily verify that, with the same definitions of addition and scalar multiplication as we had in , the set satisfies the axioms (A) , (B) , and (C) of Section: Vector spaces .

Two special examples of subspaces are:

the set consisting of the origin only, and
the whole space .

The following examples are less trivial.

Example 1. Let and be any two strictly positive integers, . Let be the set of all vectors in for which .

Example 2. With and as in (1), we consider the space , and any real numbers . Let be the set of all vectors (polynomials) in for which .

Example 3. Let be the set of all vectors in for which holds identically in .

We need some notation and some terminology. For any collection of subsets of a given set (say, for example, for a collection of subspaces in a vector space ), we write , for the intersection of all , i.e., for the set of points common to them all. Also, if and are subsets of a set, we write if is a subset of , that is, if every element of lies in also. (Observe that we do not exclude the possibility ; thus we write as well as .) For a finite collection , we shall write in place of ; in case two subspaces and are such that , we shall say that and are disjoint .

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