Let us start with a given matrix

.

The aim of this entry is to compare nullspace, column space and row space between , and . Obviously, and are and matrices respectively.

**Nullspaces**. We start with the following result

Proposition1. The followingholds.

*Proof*. Pick an arbitrary element , i.e. , we can see that

so

.

Conversely, assume is such that , as a consequence, . This gives us the fact

.

Consequently, which proves

.

In other words, .

**Remark**. is no longer true since these two matrices have different dimension.

Regarding to matrix , one has

.

Similarly, involving matrix , one gets

.

It now follows from Proposition 1 that

.

In other words, column and row spaces associated to and have the same dimension respectively.

**Row spaces**. We prove the following

Proposition2. The followingholds.

*Proof*. The way to compare column spaces is to use the following facts

and

.

Equivalently, from the first fact we need to show that

.

In term of the second fact, once you have a suitable matrix the column space of is indeed contained in the column space of . Therefore

which turns out to be

since they have the same dimension, equality occurs.

**Remark**. This comes from the proof above. If you have a good matrix , the following is true

.

**Column spaces**. We prove the following

Proposition3. The followingholds.

*Proof*. This is trivial by using Proposition 2.

**Remark**. If you have a good matrix , the following is true

.