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 \(\mathcal{M}\) of a vector space \(\mathcal{V}\) is a subspace or a linear manifold if along with every pair, \(x\) and \(y\) , of vectors contained in \(\mathcal{M}\) , every linear combination \(\alpha x+\beta y\) is also contained in \(\mathcal{M}\) .
A word of warning: along with each vector \(x\) , a subspace also contains \(x-x\) . 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 \(\mathcal{M}\) in a vector space \(\mathcal{V}\) is itself a vector space; the reader can easily verify that, with the same definitions of addition and scalar multiplication as we had in \(\mathcal{V}\) , the set satisfies the axioms (A) , (B) , and (C) of Section: Vector spaces .
Two special examples of subspaces are:
- the set \(\mathcal{O}\) consisting of the origin only, and
- the whole space \(\mathcal{V}\) .
The following examples are less trivial.
Example 1. Let \(n\) and \(m\) be any two strictly positive integers, \(m \leq n\) . Let \(\mathcal{M}\) be the set of all vectors \(x=(\xi_{1}, \ldots, \xi_{n})\) in \(\mathbb{C}^{n}\) for which \(\xi_{1}=\cdots=\xi_{m}=0\) .
Example 2. With \(m\) and \(n\) as in (1), we consider the space \(\mathcal{P}_{n}\) , and any \(m\) real numbers \(t_{1}, \ldots, t_{m}\) . Let \(\mathcal{M}\) be the set of all vectors (polynomials) \(x\) in \(\mathcal{P}_n\) for which \(x(t_{1})=\cdots=x(t_{m})=0\) .
Example 3. Let \(\mathcal{M}\) be the set of all vectors \(x\) in \(\mathcal{P}\) for which \(x(t)=x(-t)\) holds identically in \(t\) .
We need some notation and some terminology. For any collection \(\{\mathcal{M}_{\nu}\}\) of subsets of a given set (say, for example, for a collection of subspaces in a vector space \(\mathcal{V}\) ), we write \(\bigcap_{\nu} \mathcal{M}_{\nu}\) , for the intersection of all \(\mathcal{M}_{\nu}\) , i.e., for the set of points common to them all. Also, if \(\mathcal{M}\) and \(\mathcal{N}\) are subsets of a set, we write \(\mathcal{M} \subset \mathcal{N}\) if \(\mathcal{M}\) is a subset of \(\mathcal{N}\) , that is, if every element of \(\mathcal{M}\) lies in \(\mathcal{N}\) also. (Observe that we do not exclude the possibility \(\mathcal{M}=\mathcal{N}\) ; thus we write \(\mathcal{V} \subset \mathcal{V}\) as well as \(\mathcal{O} \subset \mathcal{V}\) .) For a finite collection \(\{\mathcal{M}_{1}, \ldots, \mathcal{M}_{n}\}\) , we shall write \(\mathcal{M}_{1} \cap \cdots \cap \mathcal{M}_{n}\) in place of \(\bigcap_{\nu} \mathcal{M}_{\nu}\) ; in case two subspaces \(\mathcal{M}\) and \(\mathcal{N}\) are such that \(\mathcal{M} \cap \mathcal{N}=\mathcal{O}\) , we shall say that \(\mathcal{M}\) and \(\mathcal{N}\) are disjoint .