If the systems $Ax=u$ and $Ax=v$ don't have solutions, is it possible that $Ax=u+v$ has a solution? For instance, for $u=(0,2,4,6)^T$ and $v=(1,3,5,7)^T$, is it possible that the system
$$
Ax=u+v
$$
has a solution if $Ax=u$ and $Ax=v$ have no solution? I know that if $Ax=b$ and $Ay=b$ with $x$ and $y$ solutions of the system, then $A(x-y)=0$, although this is not the case.
 A: If $Ax=u$    has no solution, it is also true that $Ax = -u$  has no solution. So we are saying  your $v=-u$
Next,
$Ax = u + v = u +(-u) = u-u=0$  always has a solution
A: Hint. If $Ax=b$ has a solution and $Ax=u$ doesn't, then $Ax=b-u$ also doesn't.
$\big[$ It's a lot like asking if the sum of irrationals can be rational. The rationals are an additive subgroup of the reals, and the irrationals are its set-theoretic complement. The set of $b$ for which $Ax=b$ has a solution is a subspace of whatever vector space you're working with, and the set of $u$ for which $Ax=u$ doesn't have a solution is its set-theoretic complement. $\big]$
A: It's possible. Let's consider this scenario: the column space of A is a plane through origin, and u is a line through origin but not on the plane, v is an another line through origin but not on the plane, so u and v satisfy $Ax=u$ and $Ax=v$ don't have solutions. But it is evident that v+u is possible to be on the plane (you can have a try on Scratch paper), so $Ax=u+v$ is possible to have a solution despite $Ax=u$ and $Ax=v$ don't have solutions. In conclusion, it has no direct connection between these equations.
A: I´ve consulted "Introduction to Linear Algebra" by Gilbert Strang and I've already solved it. According to your answers, it is posible to have a matrix $A$ with such characteristics.
In the specific case when $Ax=u$ and $Ax=v$ don't have solution, it is posible that $Ax=(u+v)$ has a solution if $u+v$ is in the columns space of $A$ while $v$ and $u$ don't. Therefore, the columns space of $A$ is made up of vector of the form c(u+v) with $c \in R$.
