Let $T:\mathbb{R^4}\rightarrow \mathbb{R^{4}}$ be defined by $T(x,y,z,w)=(x+y+5w,x+2y+w,-z+2w,5x+y+2z)$ then what would be the dimension of the eigenspace of $T$?
One approach may be to find out eigenvalues and then eigenvectors. Is there any other approach that will consume less amount of time and calculation?