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
Proposition 1. The following
Proof. Pick an arbitrary element , i.e. , we can see that
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
Proposition 2. The following
Proof. The way to compare column spaces is to use the following facts
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
Proposition 3. The following
Proof. This is trivial by using Proposition 2.
Remark. If you have a good matrix , the following is true