How to solve for and write out an eigenspace I have the matrix \begin{pmatrix}   3 & 2 \\  4 & 1 \end{pmatrix} and it has eigenvalues $\lambda = 5$ and $\lambda = -1$.
I need to find the eigenspace for each eigenvalue. I've done a calculation myself but I'm not sure if it's correct and I don't know how I should write my answer either. 
I know that using $\lambda = 5$ we get \begin{pmatrix} -2 & 2 \\ 4 & -4 \end{pmatrix} which has (I think) an eigenspace of $(1,1)$?
And with $\lambda = -1$ we get \begin{pmatrix}4 & 2 \\ 4 & 2 \end{pmatrix} which has (I think) an eigenspace of $(1, -1/2)$.
Can somebody check my work and correct me if I'm wrong, and show me the correct way to write out an eigenspace. If I did so correctly, I would love an explanation of why I did so correctly because I don't really understand myself.
Thanks.
 A: You can check your work by plugging in your solution and seeing if you get the zero vector.
For $\lambda=5$, we have 
$$\begin{pmatrix}-2&2\\4&-4\end{pmatrix}\begin{pmatrix}1\\1 \end{pmatrix}=
\begin{pmatrix}0\\0 \end{pmatrix}$$
So $\begin{pmatrix} 1\\1\end{pmatrix}$ is an eigenvector with eigenvalue $\lambda=5$.
For $\lambda = -1$, we have
$$\begin{pmatrix}4&2\\4&2 \end{pmatrix}\begin{pmatrix}1\\-1/2 \end{pmatrix}=\begin{pmatrix}3\\3 \end{pmatrix}$$
So something's not right here.  Indeed, the vector you should obtain is (a scalar multiple of) $\begin{pmatrix}1\\-2 \end{pmatrix}$.
A: Take $\lambda=5$, we need to find $\mathcal{N}(A-5I_2)=\{v\in\mathbf{R}^2\mid (A-5I_2)v=0\}$.
Therefore we use Gaussian elimination to find the set of all $v\in\mathbf{R}^2$ such that $(A-5I_2)v=0$, where $A-5I_2$ is the matrix down below.
$$\begin{bmatrix}-2 & 2 \\ 4 & -4\end{bmatrix}\sim\begin{bmatrix}1 & -
1 \\ 0 & 0 \end{bmatrix}$$
Let $x_2=\mu$, then $x_1=\mu$, so $\mathcal{N}(A-5I_2)=E_5=\{(1,1)\mu\mid \mu\in\mathbf{R}\}$. 
This is the eigenspace corresponding to $\lambda=5$ by definition.
The same procedure works for the other eigenvalue.
