How to show that if $W\subset V$ is invariant under $T:V\to V$ and $\dim{W}=1$, then $W$ is spanned by an eigenvector for $T$?

Question

$W$ is invariant if $T(W)\subset W$, meaning the result can be expressed in terms of vectors in $W$. But I don't understand how this can be related to eigenvector? Could someone help?

Answer 1

If $W$ is a one-dimensional invariant subspace, then $$W = \{cv:c\in\mathbb F\} $$ for some nonzero $v\in V$(where $\mathbb F$ is the field of scalars.). Since $Tv\in W$, it follows that $Tv=\lambda v$ for for some $\lambda\in\mathbb F$, so that $v$ is an eigenvector.

Conversely, if $v$ is an eigenvector for $T$, then it is easy to verify that $\{cv:c\in\mathbb F\}$ is a one-dimensional invariant subspace.