# I don't understand a “syntax” of vector subspace

I need to solve this question:

Decide which of these subsets are vector subspace of $\mathbb{R}^n$ ($\mathbb{R}$ is Real field):
$\{x \mid Ax = b\}$, where $A_{(m,n)} = 0$ and $b_{(m,1)} = 0$.

Could you please explain me and help me understand this question?

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What is Rn? What is 'a "syntax" of vector subspace'? –  Chris Eagle Dec 2 '12 at 20:12
Did you mean $\mathbb{R}^n$ ? $x$ an $n\times 1$ vector? –  amWhy Dec 2 '12 at 20:15

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.