Let $A$ be the matrix representing the linear transformation $f : ( x , y ) \rightarrow ( 2 x - y , x - 2 y )$. When the point $( 5 , - 1 )$ is mapped to the point $( a , b )$ by the linear transformation represented by the matrix $A ^ { 4 }$, find the value of $a + b$. [3 points]