Eigenspaces/Direct sum/Dimension/Fact/Proof

Proof

The directness means

${}\operatorname {Eig} _{\lambda _{k}}{\left(\varphi \right)}\cap {\left(\sum _{i\neq k}\operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}\right)}=0\,$

for every ${}k$. Suppose that this is not the case. Then we had vectors ${}v_{i}\in \operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}$ fulfilling

${}0\neq v_{k}=\sum _{i\neq k}v_{i}\,.$

But this contradicts fact.

To fact