Solution
= Solution
We use induction on $d$. Suppose
$$
\sum_{i=1}^dc_iw_i=0.
$$
Applying $f-\alpha_dI$ gives
$$
\sum_{i=1}^{d-1}c_i(\alpha_i-\alpha_d)w_i=0.
$$
By induction, every $c_i(\alpha_i-\alpha_d)=0$ for $i<d$. The eigenvalues are distinct, so $c_i=0$ for $i<d$, and the original relation then gives $c_d=0$. Thus eigenvectors with distinct eigenvalues are <linear independence>[linearly independent].