It is difficult to figure out what you are asking. So I assume that you have the following: $\mathbb{R}$ is the real numbers. You have the vector space $\mathbb{R}^n$ of dimension $n$. You then have a subset of $\mathbb{R}^n$ $ Y = \{x\in \mathbb{R}^n : Ax = b\} $ where $A$ is a fixed $n\times n$ matrix with entries from $\mathbb{R}$ and $b\in \mathbb{R}^n$. You assume that the matrix $A$ has the $(m,n)$th entry zero and that the $m$th coordinate of $b$ is also zero.
Now if you have $Ax = b$ and $Ay = b$, then $A(x+y) = Ax + Ay = b + b = 2b$. This means that $b$ has to be ... for $Y$ to be a vector space.