Matrix notation $i$ $j$ Let $A = \begin{bmatrix}
    a_1       & a_2 & \cdots & a_n \\
    \end{bmatrix}$ be a $n \times n$ matrix such that $a_i \cdot a_i = 1$ for all $i$ and $a_i \cdot a_j = 0$ for all $i \neq j$. 
I'm familiar with $i$ indicating row and $j$ indicating column but I'm not sure what these dot products actually refer to.
Let's say we have $B = \begin{bmatrix}
    \frac{1}{\sqrt2}       & \frac{1}{\sqrt2}\\
    -\frac{1}{\sqrt2} & \frac{1}{\sqrt2}\\ \end{bmatrix}$.
What does $a_i \cdot a_i =1$ and $a_i \cdot a_j = 0$ mean here?

Such $B$ doesn't fulfill your constraints. Perhaps you intended$$B=\begin{bmatrix}\dfrac{1}{\sqrt 2}&-\dfrac{1}{\sqrt 2}\\\dfrac{1}{\sqrt 2}&\dfrac{1}{\sqrt 2}\end{bmatrix}$$here we define $$a_m=\begin{bmatrix}b_{1m}\\b_{2m}\\.\\.\\.\\b_{nm}\end{bmatrix}\qquad,\qquad \forall m$$and $$a_i\cdot a_j=\sum_{k=1}^{n}b_{ki}b_{kj}$$
