Let $V$ be a [vector space](/page/Vector%20Space) over a field $k$, let $T:V\to V$ be a [linear map](/page/Linear%20Map), and let $L\subset V$ be a one-dimensional subspace. Then $L$ is invariant under $T$, meaning $T(L)\subset L$, if and only if there exists a scalar $\lambda\in k$ such that every nonzero vector $w\in L$ satisfies

\begin{align*} T(w)=\lambda w. \end{align*}

Equivalently, every nonzero vector in $L$ is an eigenvector of $T$ with the same eigenvalue $\lambda$.