If $A$ if of full column rank, what about $[A;B]$

Question

I meet a fundamental problem as the following:

Suppose $A$ is full column rank, is $$C = \begin{bmatrix}A \\ B\end{bmatrix}$$ also full column rank? Suppose $B$ has the same number of column as $A$.

I think the answer is yes since $B$ does not influence the independence of columns of $C$ if $A$ has linearly independent columns.

thanks in advance.

Answer 1

You are right, $B$ does not influence the independence of columns of $C$

One way to see that: $C$ is full column rank iff $x\mapsto Cx$ is injective (one-to-one mapping).

To prove that $x\mapsto Cx$ is injective one must show that $Cx=0\Rightarrow x=0$. But we have $\left(\begin{array}{c} 0 \\ 0\end{array}\right)=\left(\begin{array}{c} Ax \\ Bx\end{array}\right)$ with $A$ full column rank (id injective) hence $x=0$, hence $C$ is full column rank.