I am self studying Linear Algebra from Textbook Hoffman and Kunze and I have a question in Theorem 10 of Chapter 9 .
Adding it's image:
How it's clear that for every $\alpha$ in $E_{j}V $ f(T) $\alpha$ = f($c_{j} $) $\alpha$ ? Can anyone please tell reasoning behind it.
Best Answer
If $\alpha\in E_jV$, then $\alpha=E_jv$ for some vector $v$. Therefore $$ f(T)\alpha=\sum_if(c_i)E_iE_jv=f(c_j)E_j^2v=f(c_j)\alpha. $$