How to determine if a set of five $2\times2$ matrices is independent $$S=\bigg\{\left[\begin{matrix}1&2\\2&1\end{matrix}\right], \left[\begin{matrix}2&1\\-1&2\end{matrix}\right], \left[\begin{matrix}0&1\\1&2\end{matrix}\right],\left[\begin{matrix}1&0\\1&1\end{matrix}\right], 
\left[\begin{matrix}1&4\\0&3\end{matrix}\right]\bigg\}$$
How can I determine if a set of five $2\times2$ matrices are independent? 

As the others have said, this set of $5$ must be linearly dependent because the dimension of the space of all $2\times 2$ matrices is $4$.  
More generally, how do you show that a set of vectors is linearly dependent or independent?  Create a linear combination of the vectors, set it equal to $0$, and try to solve it.  
$$
a_1X_1 + a_2X_2 + \dotsb+ a_nX_n = 0
$$
If the only possible solution is $a_1 = a_2 = \dotsb = a_n = 0$ then the set is independent.  If a different solution exists then the set is dependent.
