Column Space, Rank and Matrix Concatenation

Question

I have the following question:

Given Matrices $A$ and $B$, the following relation exists between their column spaces:

$$\text{col}(B) \subseteq \text{col}(A)$$

Then, which of the following is true for Matrix $C=[A\,\,\,\,\,B]$?

A) $\text{rank}(C)=\text{rank}(A)$

B) $\text{rank}(C)=\text{rank}(B)$

C) It is not possible to specify $\text{rank}(C)$ in terms of $\text{rank}(A)$ and $\text{rank}(B)$

My guess, and it seems a reasonable one, would be alternative (A), but I don't know how to solve/express it mathematically.

Is my guess correct? Could you walk me through the steps to prove it?

Thanks in advance

Answer 1

Yes, that's correct.

The columns of $C$ are just the columns of $A$ followed by the columns of $B$, which are all included in $\mathrm{col}(A)$, hence $$\mathrm{col}(C) \ =\ \mathrm{col}(A)$$ so their dimensions - the ranks - coincide.