Finding Dual Basis from a basis in $\mathbb R^2$ I am given the fact that these two vectors form the basis B of $\mathbb R^2$:
$$
B=\{\begin{bmatrix} 2 \\ 1 \\ \end{bmatrix} \begin{bmatrix} 3 \\ 1 \end{bmatrix}\}
$$
and then asked to find the dual basis or the basis of $(\mathbb R^2)^*$. 
I would really appreciate if anyone could explain the procedure to follow for these types of questions.

You want a pair of vectors $v^1,v^2$ such that $v^iv_j=\delta^i_j=\begin{cases}1, i=j\\0, i\neq j\end{cases}$.
So, $v^1=(-1,3),v^2=(1,-2)$ would work.
For $v^1$ you get the system $\begin{cases}  2x+y=1\\3x+y=0\end{cases}$, and similarly for $v^2$.
