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 which 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 to find a spanning set of vectors.

Suppose that is a nonempty 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 “nonemptiness”.

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.

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 order to do computations, however, it is usually necessary to find a spanning set.

To be clear: the null space 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

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