# Column Space, Rank and Matrix Concatenation

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?

• It is true that $\mathrm{rank}(C) = \mathrm{rank}(A)$, since $\mathrm{col}(C)\subset\mathrm{col}(A)$ and vice versa. – Math1000 Jan 4 at 22:19
• $C \binom{x}{y} = Ax + By$. Hence ${\cal R}C = {\cal R}A + {\cal R} B$. You are given ${\cal R}B \subset {\cal R} A$. – copper.hat Jan 4 at 22:31
• Is it correct that if $\text{col}(B) \subseteq \text{col}(A)$, then $\text{rank}(B) \leq \text{rank}(A)$ and the rank of a concatenated matrix is equal to the maximum rank among the concatenating matrices? Is this a way to solve it? – bertozzijr Jan 4 at 23:11
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.