Range and null-space

Definition 1. If is a linear transformation on a vector space and if is a subspace of , the image of under , in symbols , is the set of all vectors of the form with in . The range of is the set ; the null-space of is the set of all vectors for which .

It is immediately verified that and are subspaces. If, as usual, we denote by the subspace containing the vector only, it is easy to describe some familiar concepts in terms of the terminology just introduced; we list some of the results.

The transformation is invertible if and only if and .
In case is finite-dimensional, is invertible if and only if or .
The subspace is invariant under if and only if .
A pair of complementary subspaces and reduce if and only if and .
If is the projection on along , then and .

All these statements are easy to prove; we indicate the proof of (v). From Section: Projections , Theorem 2, we know that is the set of all solutions of the equation ; this coincides with our definition of . We know also that is the set of all solutions of the equation . If is in , then is also in , since is the image under of something (namely of itself). Conversely, if a vector is the image under of something, say, (so that is in ), then , so that is in .

Warning: it is accidental that for projections . In general it need not even be true that and are disjoint. It can happen, for example, that for a certain vector we have , , and ; for such a vector, clearly belongs to both the range and the null-space of .

Theorem 1. If is a linear transformation on a vector space , then if is finite-dimensional, then

Proof. If is in , then, for all in , so that and is in . If, on the other hand, is in , then, for all in , so that is in .

If we apply (1) to in place of , we obtain If is finite-dimensional (and hence reflexive), we may replace by in (3), and then we may form the annihilator of both sides; the desired conclusion (2) follows from Section: Annihilators , Theorem 2. ◻

EXERCISES

Exercise 1. Use the differentiation operator on to show that the range and the null-space of a linear transformation need not be disjoint.

Exercise 2.

Give an example of a linear transformation on a three-dimensional space with a two-dimensional range.
Give an example of a linear transformation on a three-dimensional space with a two-dimensional null-space.

Exercise 3. Find a four-by-four matrix whose range is spanned by and .

Exercise 4.

Two projections and have the same range if and only if and
Two projections and have the same null-space if and only if and .

Exercise 5. If are projections with the same range and if are scalars such that , then is a projection.

EXERCISES

Quick Links

Social Media