This following problem is a property of column space in my Textbook which I am trying to prove.
Problem: For any $A$ and $B$, $C(AB) \subseteq C(A)$.
My approach:
Let us consider $A \in R^{m \times n}$, $B \in R^{n \times m}$, and $AB \in R^{m \times m}$.
If $z \in C(AB)$, then $z=AB x $, where $x \in R^m$.
Let $y \in C(B)$, then $y=B x $, where $x \in R^m$.
This implies $z = ABx = Ay$, where $y \in R^n$ and thus $z \in C(A)$
From the above argument, we can write $C(AB) \subset C(A)$.
My question:
- How can I prove the subset and equality?
- If possible, please explain the notion of subset, subset and equality in regards to vector space. That would be really helpful.
Note: This is not an exam/assignment question.