Discover Linear Algebra

Subsection 17.3.1 Recognizing subspaces

How can we recognize subspaces? To be a subspace, a subcollection of vectors must satisfy all ten vector space axioms on its own. But Axiom A 2, Axiom A 3, and Axioms S 2–5 all concern the algebra of vectors, and don't really take into account where the vectors are considered to “live”. Since these algebra axioms are true about all vectors in the larger space, they are automatically true about the vectors in the subcollection. So that leaves Axiom A 1, Axiom A 4, Axiom A 5, and Axiom S 1.

For axioms Axiom A 1 and Axiom S 1, we already have addition and scalar multiplication of vectors defined as in the large space. But when we are considering whether the smaller collection is a vector space all on its own, the vectors not in this collection are no longer relevant. So the parts of these axioms that say “the result is always equal to another in the collection of objects” now refer to the subcollection of vectors under consideration. That is, we need to make sure that when vectors in the subcollection are added or scalar multiplied, then the result is again in the subcollection, not somewhere else in the vector space at large. We call this property being closed under the vector operations.

Similarly, for Axiom A 5, we already know that every vector in the large space has a negative, hence so does every vector in the subcollection. But we need to check that the subcollection is closed under taking negatives — that the negative of a vector in the subcollection is again in the subcollection. And we know that there is a zero vector in the large space, but the subcollection needs a zero vector too, to satisfy Axiom A 4. The zero vector from the larger space already satisfies the property that \(\uvec{v}+\zerovec=\uvec{v}\) for all vectors, but again we need the zero vector to be in the subcollection.

As we will prove in Proposition 17.5.1, we really only need to check a subcollection for closure under addition and scalar multiplication in order to verify that it is also a vector space.

See Section 17.4 for examples of applying the Subspace Test to verify that certain subcollections of vectors in vector spaces form subspaces.

Subsection 17.3.2 Building subspaces

Suppose we are studying a problem for which certain vectors in a certain vector space are important. We would like to do linear algebra with these certain special vectors, so the fact that they are part of a vector space is essential, but perhaps not all of the vector space in which they live is relevant to the problem. Can we find a smaller subspace which contains these important vectors? Even better, can we determine the smallest subspace which contains these important vectors?

As stated above, every subspace must satisfy the three parts of the Subspace Test. So if a subspace contains our special vectors, then it must also contain all scalar multiples of those vectors. And it must also be closed under vector addition, so it must also contain all sums of scalar multiples of the special vectors. Therefore, it must contain every linear combination of the special vectors. (In other words, subspaces are also closed under taking linear combinations.)

As we noted in Discovery guide 17.1, and will verify in Subsection 17.5.2 (Proposition 17.5.5), the subcollection of a vector space consisting of all linear combinations of a set of vectors \(S\) is always a subspace, called the span of \(S\) and written \(\Span S\text{.}\) So the process of taking all possible linear combinations of a set of vectors can be used to build subspaces. And sometimes, as in Discovery 17.4, the space that span builds ends up being the whole larger space.

Subsection 17.3.3 The subspaces of \(\R^n\)

We saw in Subsection 15.3.2 that a line through the origin in \(\R^n\) can be described in vector form as \(\uvec{x} = t \uvec{p}\) for some vector \(\uvec{p}\) that is parallel to the line (and taking “initial” point \(\uvec{x}_0 = \zerovec\text{,}\) since the line passes through the origin). Similarly, we saw that a plane through the origin in \(\R^3\) can be described in vector form as \(\uvec{x} = s \uvec{p}_1 + t \uvec{p}_2\) for some vectors \(\uvec{p}_1\) and \(\uvec{p}_2\) that are parallel to the plane but not parallel to each other. With our new, more sophisticated view of vector spaces and subspaces, we can now recognize a line \(\uvec{x} = t \uvec{p}\) as the subspace \(\Span \{\uvec{p}\}\text{,}\) and a plane \(\uvec{x} = s \uvec{p}_1 + t \uvec{p}_2\) as the subspace \(\Span \{\uvec{p}_1,\uvec{p}_2\}\text{.}\)

In fact, every (nontrivial) subspace of \(\R^n\) has a geometric interpretation as a line or a plane or some sort of higher-dimensional hyperplane. In particular,

• the subspaces of \(\R^2\) are precisely

  • the zero space \(\{\zerovec\}\text{,}\)
  • \(\Span \{\uvec{p}\}\) for a nonzero vector \(\uvec{p}\text{,}\) which builds a line through the origin, and
  • \(\Span \{\uvec{p}_1,\uvec{p}_2\}\) for two nonzero, nonparallel vectors \(\uvec{p}_1,\uvec{p}_2\text{,}\) which builds the whole plane \(\R^2\text{;}\)

• the subspaces of \(\R^3\) are precisely

  • the zero space \(\{\zerovec\}\text{,}\)
  • \(\Span \{\uvec{p}\}\) for a nonzero vector \(\uvec{p}\text{,}\) which builds a line through the origin,
  • \(\Span \{\uvec{p}_1,\uvec{p}_2\}\) for two nonzero, nonparallel vectors \(\uvec{p}_1,\uvec{p}_2\text{,}\) which builds a plane through the origin, and
  • \(\Span \{\uvec{p}_1,\uvec{p}_2,\uvec{p}_3\}\) for three nonzero, non-coplanar vectors \(\uvec{p}_1,\uvec{p}_2,\uvec{p}_3\text{,}\) which builds all of space \(\R^3\text{;}\)

• the subspaces of \(\R^4\) are precisely

  • the zero space \(\{\zerovec\}\text{,}\)
  • \(\Span \{\uvec{p}\}\) for a nonzero vector \(\uvec{p}\text{,}\) which builds a line through the origin,
  • \(\Span \{\uvec{p}_1,\uvec{p}_2\}\) for two nonzero, nonparallel vectors \(\uvec{p}_1,\uvec{p}_2\text{,}\) which builds a plane through the origin,
  • \(\Span \{\uvec{p}_1,\uvec{p}_2,\uvec{p}_3\}\) for three nonzero, non-coplanar vectors \(\uvec{p}_1,\uvec{p}_2,\uvec{p}_3\text{,}\) which builds a hyperplane through the origin, and
  • \(\Span \{\uvec{p}_1,\uvec{p}_2,\uvec{p}_3,\uvec{p}_4\}\) for four nonzero, non-cohyperplanar vectors \(\uvec{p}_1,\uvec{p}_2,\uvec{p}_3,\uvec{p}_4\text{,}\) which builds all of four-dimensional space \(\R^4\text{;}\)
• etc.

Subsection 17.3.4 Recognizing when two subspaces are the same

One of the goals of the next few chapters is to determine how to describe vector spaces using spanning sets in an optimal fashion. In this endeavour, we will want to know when a refinement of a spanning set still spans the space we are trying to describe. So, in particular, we will need to know when two spanning sets generate the same subspace. Since spans are defined by linear combinations, and subspaces are closed under linear combinations, this will not be as difficult as it sounds, and we provide a test for this situation as Proposition 17.5.6 in Subsection 17.5.3.