Explanation in some part of proof It might be too specific to be asked here, but I can't figure out myself. I have this thm and my problem is in the necessity part:
$$
\text{M is hyperplane iff } \exists h\in\mathbf{R}^n \text{ and } r\in\mathbf{R} \text{ such that } M=\left\{x\in\mathbf{R}^n\mid h.x=r   \right\}
$$
My definition for hyperplane is:
$$
\text{ M is hyperplane iff } M=\left\{m\right\}+W
$$ 
where $m$ is a point and $W$ is a subspace with dimension $n-1$. Given this definition, it is straightforward to say, $\exists \bar{x}\in\mathbf{R}^n\setminus W$. Then, a linear map $h:\mathbf{R}^n\to\mathbf{R}$ is defined so that $h(x)=0,x\in W$, $h(\bar{x})=1$ and $h(m)=r$. Then it is said it is straightforward to check the claim.
My question is: $h$ takes vector and gives a scalar then what is the meaning of $h.x$? It looks we are multiplying a vector with a scalar, but then $r$ itself is a scalar. So, I could not figure out how this proof works. Any help would be greatly appreciated!

The $\cdot$ on the right hand side represents the inner product: $h \cdot x = \sum_{i=1}^n h_i x_i$. This means that you also have to revise your proof of $\Rightarrow$. Furthermore, in the statement, $h$ should be required to be $\neq 0$, otherwise (with $h = 0$ and $r = 0$) ${\mathbb R}^n$ also satisfies the right hand side, but it is not a hyperplane.
