Let $V$ be a possibly infinite vector space. Let $T:V \rightarrow V$ and $F:V \rightarrow V$ be two isomorphic linear transformations. Then, is $T + F$ also an isomorphism?
I believe the answer is yes. Since $T$ and $F$ are both automorphisms, there should be no reason as to why their sum would not be an isomorphism as well, right? My reasoning is as follows:
Let $v \in V$. Then $T + F$ has the mapping $T(v) + F(v)$, and this is a linear transformation, since $\forall$ real scalars $\alpha$, $\beta$ and $\forall \thinspace \thinspace v,w \in V$:
$\begin{align} (T+F)(\alpha v + \beta w) = & \enspace T(\alpha v + \beta w) +F(\alpha v + \beta w) \\ = & \enspace \alpha T(v) + \beta T(w) + \alpha F(v) + \beta F(w) \\ = & \enspace \alpha(T + F)(v) + \beta (T+F)(w) \end{align}$
which itself is an isomorphism. Is this all correct?
How about $T=id$ and $F=-id$?