In this section we discuss subspaces of A subspace turns out to be exactly the same thing as a span, except we don’t have a particular set of spanning vectors in mind. This change in perspective is quite useful, as it is easy to produce subspaces that are not obviously spans. For example, the solution set of the equation is a span because the equation is homogeneous, but we would have to compute the parametric vector form in order to write it as a span.

(A subspace also turns out to be the same thing as the solution set of a homogeneous system of equations.)

Subsection2.6.1Subspaces: Definition and Examples

Definition

A subset of is any collection of points of

For instance, the unit circle

is a subset of

Above we expressed in set builder notation: in English, it reads “ is the set of all ordered pairs in such that ”

Definition

A subspace of is a subset of satisfying:

Non-emptiness: The zero vector is in

Closure under addition: If and are in then is also in

Closure under scalar multiplication: If is in and is in then is also in

As a consequence of these properties, we see:

If is a vector in then all scalar multiples of are in by the third property. In other words the line through any nonzero vector in is also contained in

If are vectors in and are scalars, then are also in by the third property, so is in by the second property. Therefore, all of is contained in

Similarly, if are all in then is contained in In other words, a subspace contains the span of any vectors in it.

If you choose enough vectors, then eventually their span will fill up so we already see that a subspace is a span. See this theorem below for a precise statement.

Suppose that is a non-empty subset of that satisfies properties 2 and 3. Let be any vector in Then is in by the third property, so automatically satisfies property 1. It follows that the only subset of that satisfies properties 2 and 3 but not property 1 is the empty subset This is why we call the first property “non-emptiness”.

Example

The set is a subspace of itself: indeed, it contains zero, and is closed under addition and scalar multiplication.

Example

The set containing only the zero vector is a subspace of it contains zero, and if you add zero to itself or multiply it by a scalar, you always get zero.

A subset of is any collection of vectors whatsoever. For instance, the unit circle

is a subset of but it is not a subspace. In fact, all of the non-examples above are still subsets of A subspace is a subset that happens to satisfy the three additional defining properties.

In order to verify that a subset of is in fact a subspace, one has to check the three defining properties. That is, unless the subset has already been verified to be a subspace: see this important note below.

To show that is a subspace, we have to verify the three defining properties.

The zero vector is in the span.

If and are in then
is also in

If is in and is a scalar, then
is also in

Since satisfies the three defining properties of a subspace, it is a subspace.

Now let be a subspace of If is the zero subspace, then it is the span of the empty set, so we may assume is nonzero. Choose a nonzero vector in If then we are done. Otherwise, there exists a vector that is in but not in Then is contained in and by the increasing span criterion in Section 2.5, the set is linearly independent. If then we are done. Otherwise, we continue in this fashion until we have written for some linearly independent set This process terminates after at most steps by this important note in Section 2.5.

If we say that is the subspace spanned by or generated by the vectors We call a spanning set for

Any matrix naturally gives rise to two subspaces.

Definition

Let be an matrix.

The column space of is the subspace of spanned by the columns of It is written

The null space of is the subspace of consisting of all solutions of the homogeneous equation

The column space is defined to be a span, so it is a subspace by the above theorem. We need to verify that the null space is really a subspace. In Section 2.4 we already saw that the set of solutions of is always a span, so the fact that the null spaces is a subspace should not come as a surprise.

The column space and the null space of a matrix are both subspaces, so they are both spans. The column space of a matrix is defined to be the span of the columns of The null space is defined to be the solution set of so this is a good example of a kind of subspace that we can define without any spanning set in mind. In other words, it is easier to show that the null space is a subspace than to show it is a span—see the proof above. In order to do computations, however, it is usually necessary to find a spanning set.

Null Spaces are Solution Sets

The null space of a matrix is the solution set of a homogeneous system of equations. For example, the null space of the matrix

is the solution set of i.e., the solution set of the system of equations

Conversely, the solution set of any homogeneous system of equations is precisely the null space of the corresponding coefficient matrix.

To find a spanning set for the null space, one has to solve a system of homogeneous equations.

Recipe: Compute a spanning set for a null space

To find a spanning set for compute the parametric vector form of the solutions to the homogeneous equation The vectors attached to the free variables form a spanning set for

Writing a subspace as a column space or a null space

A subspace can be given to you in many different forms. In practice, computations involving subspaces are much easier if your subspace is the column space or null space of a matrix. The simplest example of such a computation is finding a spanning set: a column space is by definition the span of the columns of a matrix, and we showed above how to compute a spanning set for a null space using parametric vector form. For this reason, it is useful to rewrite a subspace as a column space or a null space before trying to answer questions about it.

When asking questions about a subspace, it is usually best to rewrite the subspace as a column space or a null space.

This also applies to the question “is my subset a subspace?” If your subset is a column space or null space of a matrix, then the answer is yes.

Example

Let

be the subset of a previous example. The subset is exactly the solution set of the homogeneous equation Therefore,

In particular, it is a subspace. The reduced row echelon form of is so the parametric form of is so the parametric vector form is and hence spans