defining subspace in $\mathbb{R}^n$ that separates two convex sets

Question

I am looking at the "separating hyperplane theorem", that I've restated below, and I was having trouble understanding the way they described the hyperplane $a^T x$. From what I understand, a is a vector in $\mathbb{R}^n$ but why take the transpose, and why multiply it into $x$?

The theorem and its proof are found here.

Thanks in advance

Let $C$ and $D$ be two convex sets in $\mathbb{R}^n$ that do not intersect. Then, there exists $a\in \mathbb{R}^n$ , $a\neq 0$, $b\in \mathbb{R}$ , such that $a^T x \leq b$ for all $x\in C$ and $a^T x \geq b$ for all $x\in D$

Answer 1

$a$ and $x$ are both column vectors.

$$a^Tx = \sum_{i=1}^n a_ix_i$$

For example $a=\begin{bmatrix} 3 \\ 2 \end{bmatrix}$, $x=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$

$$a^Tx =\begin{bmatrix} 3 & 2 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} =3x_1+2x_2$$

$3x_1+2x_2=5$ is a hyperplane.

$3x_1+2x_2 \leq 5$ is a halfspace.