Find eigenvalues and eigenspaces for linear transformation (rotation). (I) Let $T_\theta$ be a linear transformation $T_\theta : \Bbb R^2 → \Bbb R^2$  which rotates the plane $\theta$ degree anti-clockwise. For which values of $\theta$ does $T_\theta$ have at least an eigenvalue?
(II) In those cases, please find the eigenvalues and the dimension of the associated eigenspaces.
Part (I) of the question I think I've already solved correctly, but I've included it here just in case I missed something. I believe I'm only stuck on the very last bit of finding eigenspaces. Any help is greatly appreciated!
My attempt
(I): So, the rotation matrix is $$A=\begin{bmatrix}\cos \theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}$$ and in particular, given that $T\mathbf{v} = \lambda\mathbf{v}$ , we have $\det(A-\lambda I) = 0$ where $I$ is the identity matrix. 
This shows that $$det\bigg(\begin{bmatrix}\cos\theta-\lambda & -\sin\theta\\ \sin\theta & \cos\theta-\lambda\end{bmatrix}\bigg)=(\cos\theta-\lambda)^2 + \sin^2\theta=0,$$ which solving for $\theta$ yields $\theta=0$ and $\theta=\pi$ that have at least one eigenvalue. 
(II): Plugging in $\theta=0$ and $\theta=\pi$ to the above equation yields $$(\cos 0-\lambda)^2+ \sin^20=0 \qquad\implies\qquad \lambda=1$$ 
and 
$$(\cos\pi-\lambda)^2+\sin^2\pi=0 \qquad\implies\qquad \lambda=-1.$$
Now, when i try to solve for the eigenspaces, I have $T\mathbf{v} = \lambda\mathbf{v}$, so for $\theta=0$ and $\lambda = 1$, 
$$\begin{bmatrix}1-\lambda & 0\\0 & 1-\lambda\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}
= \begin{bmatrix}0 & 0\\0 & 0\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$$
A similar situation happens for when $\theta=\pi$ and $\lambda=-1$ with the zero vector. This doesn't make any sense to me as an answer. What have I done wrong, or am I just not understanding what this answer is saying?

A vector is "Eigen" when its image is parallel to itself.
This only occurs for the null rotation ($\theta=0, Rv=v$) or the reflection ($\theta=\pi, Rv=-v$). In these cases, any vector is "Eigen".
