Let $V$ a vector space of finite dimension and $T:V\rightarrow V$ a linear operator. if $W$ is subset of $V$ such that $T(W)\subset W$ and $S=T|_W$ in other words ($S:W\rightarrow W $ is defined by $S(w)=T(w))$ is the restriction of $T$ to $W$ prove $P_S(x)$ divide $P_T(x)$
I don't have idea of how to prove this exercise. I'm very stuck. Can someone help me with a hint or something?
Thanks for his time!
Consider a basis $(e_1,...,e_m)$ of $W$ that you complete with a basis of $V$, $(e_1,..,e_m,e_{m+1},..,e_n)$. The matrix of $T$ in this basis as the form:
$M_T=\pmatrix{M_S&A\cr 0&B}$, when you compute $det(xI-M_T)=det\pmatrix{xI_W-M_S&A\cr 0&B'}$ you obtain $P_S.Q$