Exercise 5.2.22

By the definition of the left hand side, it is the sum of each eigenspaces. Let $W={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{i=2}^k{E}_{{\lambda }_i}$. If there is some nonzero vector $v_1$ in $E_{{\lambda }_1}\cap W$. We may write

for some scalar $c_i$ and some eigenvectors $v_i\in E_{{\lambda }_i}$. Now we know that

After subtracting this equality by ${\lambda }_1$ times the previous equality, we get

This is impossible since ${\lambda }_i-{\lambda }_1$ is nonzero for all $i$ and $c_i$ cannot be all zero. Similarly we know that $E_{{\lambda }_i}$ has no common element other than zero with the summation of other eigenspaces. So the left hand side is the direct sum of each eigenspaces.