Image spanned by its eigenvectos

[yanky]

Here's the question I'm stuck at:
Suppose a mapping from R^2 -> R^2 is defined by some 2x2 matrix W. Is the image of W spanned by the eigenvectors of W? Why or why not?

The Attempt at a Solution

I know that the eigenvectos of W for a linearly independent set. I also know that the set spanning W will consist of the smallest subspace of W consisting of linear combinations of all vectors in W. But I'm confused about what the eigenvectors actually are. I'm assuming that the answer is "yes", but I don't know why.

 

[HallsofIvy]

Suppose
$$W= \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}$$
Then all eigenvectors or W are multiples of
$$\begin{bmatrix}0 \\ 1\end{bmatrix}$$
but the image of W is all or R².