Subspaces of Rn and Linear Independence

One of the examples that led us to introduce the idea of a vector space was the solution set of a homogeneous system. For instance, we've seen in Example 1.4 such a space that is a planar subset of $\mathbb {R} ^{3}$. There, the vector space $\mathbb {R} ^{3}$ contains inside it another vector space, the plane.

Any vector space has a trivial subspace $\{{\vec {0}}\,\}$. At the opposite extreme, any vector space has itself for a subspace. Template:AnchorThese two are the improper subspaces. Template:AnchorOther subspaces are proper.

The next result says that Example 2.8 is prototypical. The only way that a subset can fail to be a subspace (if it is nonempty and the inherited operations are used) is if it isn't closed.

Template:TextBox Briefly, the way that a subset gets to be a subspace is by being closed under linear combinations.

We usually show that a subset is a subspace with $(2)\implies (1)$.

Parametrization is an easy technique, but it is important. We shall use it often.

Template:TextBox No notation for the span is completely standard. The square brackets used here are common, but so are "${\mbox{span}}(S)$" and "${\mbox{sp}}(S)$".

The converse of the lemma holds: any subspace is the span of some set, because a subspace is obviously the span of the set of its members. Thus a subset of a vector space is a subspace if and only if it is a span. This fits the intuition that a good way to think of a vector space is as a collection in which linear combinations are sensible.

Taken together, Lemma 2.9 and Lemma 2.15 show that the span of a subset $S$ of a vector space is the smallest subspace containing all the members of $S$.

Since spans are subspaces, and we know that a good way to understand a subspace is to parametrize its description, we can try to understand a set's span in that way.

So far in this chapter we have seen that to study the properties of linear combinations, the right setting is a collection that is closed under these combinations. In the first subsection we introduced such collections, vector spaces, and we saw a great variety of examples. In this subsection we saw still more spaces, ones that happen to be subspaces of others. In all of the variety we've seen a commonality. Example 2.19 above brings it out: vector spaces and subspaces are best understood as a span, and especially as a span of a small number of vectors. The next section studies spanning sets that are minimal.