Non square linear system with two unknowns

Question

I want to know under what conditions exists at least one solution for the following system $$Ax + By = C$$ where $A$ is $p \times q$ matrix, $B$ is $p \times q$ matrix and $C$ is a vector of size $p$. Uniqueness doesn't matter. Thanks.

Answer 1

The system has a solution if and only if the system

$$\begin{bmatrix}A& B\end{bmatrix}z=C$$ has a solution. Of course, the solution vector $z$ will have $2q$ entries, and if written as

$$z=\begin{bmatrix}x\\y\end{bmatrix}$$ then $z$ is a solution of the second system if and only if $x, y$ are solutions to the original system.

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You can interpret this result by looking at $\{Ax|x\in\mathbb R^p\}$ as "the set of all linear combinations of columns of $A$". Then, $By$ is the general element of the set of all linear combinations of columns from $B$, which means that $Ax+By$ can be any linear combination of columns from both $A$ and $B$.