If $v$ is an eigenvector of operators $S$ and $T$, then $v$ is also an eigenvector of $aS + bT$ , $a, b \in F$
I know that if I let $ v_1,\space v_2 \in V$. By definition of an eigenvector, $T(v_1) = \lambda_1 v_1$, $S(v_2) = \lambda_2 v_2$,
But should I define $\lambda_1, \lambda_2$ as $a,b$, respectively instead? I'm also a bit confused about adding the vectors, as if it were under the same operator, I could have vectors $ v_1 + v_2$ simply having the linear operator $T(v_1 + v_2)$ = $T(v_1) + T(v_2)$, which proves. So I'm assuming a similar process, but I'm not exactly sure.