We now define the notion of an eigenvalue and eigenvector of a linear transformation. It should not be too surprising, especially if you remind yourself of the close relationship between matrices and linear transformations.
Suppose that $\ltdefn{T}{V}{V}$ is a linear transformation. Then a nonzero vector $\vect{v}\in V$ is an eigenvector of $T$ for the eigenvalue $\lambda$ if $\lt{T}{\vect{v}}=\lambda\vect{v}$.
We will see shortly the best method for computing the eigenvalues and eigenvectors of a linear transformation, but for now, here are some examples to verify that such things really do exist.
Here is another.
Of course, these examples are meant only to illustrate the definition of eigenvectors and eigenvalues for linear transformations, and therefore beg the question, “How would I find eigenvectors?” We will have an answer before we finish this section. We need one more construction first.