1. another question about eigenvalues and linear operators

hi, i have another question, last one - so i hope you won't mind:

i need to prove that:

S and T are two linear operators that can switch sides (ST = TS) over a vector space V.
lambda is a eigenvalue of S. the group W is defined as Ker ((lambda)I - S).
prove that W is contained in T(W).

thanks in advance to anyone who can help.

2. Originally Posted by vonflex1
hi, i have another question, last one - so i hope you won't mind:

i need to prove that:

S and T are two linear operators that can switch sides (ST = TS) over a vector space V.
lambda is a eigenvalue of S. the group W is defined as Ker ((lambda)I - S).
prove that W is contained in T(W).

thanks in advance to anyone who can help.
are you sure it's not the other way round, i.e. $\displaystyle T(W) \subseteq W$? that's a trivial result: if $\displaystyle w \in W,$ then $\displaystyle S(w)=\lambda w$ and so $\displaystyle ST(w)=TS(w)=\lambda T(w),$ which means $\displaystyle T(w) \in W.$ so $\displaystyle T(W) \subseteq W.$