Stuck on a System of linear ODE and understanding the situation. I am working on this problem in DFQ and feel as I am not understanding the concept very well. We need to solve $x'=Ax$ where we have that $x'=(x_1',x_2')$, $x=(x_1,x_2)$ and 
\begin{bmatrix}
    2       & 1  \\
    0       & 2  
\end{bmatrix}
After we find the solution we need to find the fundamental matrix.
My work so far has been to find the eigenvalues which is only one repeated $\lambda =2$ then I am stuck because of the eigenvectors. In general use usually use the relation from the matrix after we plug in the eigenvalue to find the associated eigenvector, but here we have 
\begin{bmatrix}
    0       & 1  \\
    0       & 0  
\end{bmatrix}
I would greatly appreciate if someone can help me here.
 A: The matrix:
$$\begin{bmatrix}
    2       & 1  \\
    0       & 2  
\end{bmatrix}$$
is call a Jordan Normal Form or a Jordan Block and it is not diagonalizable. This is sometimes called a deficient matrix. Have you learned the terms algebraic and geometric multiplicity? Have you learned the term generalized eigevector? In these cases, you need to find a generalized eigenvector.
In your specific case, we find a first eigenvector $[A-\lambda I]v_1 = 0$, so:
$$[A-2 I]v_1 = \begin{bmatrix}
    0       & 1  \\
    0       & 0  
\end{bmatrix}v_1 = 0 \implies v_1 = \begin{bmatrix} 1  \\  0 \end{bmatrix}$$
Because we have a deficient matrix, you cannot find a second linearly independent eigenvector, so one approach to finding a generalized eigenvector may be by solving $[A - \lambda I]v_2 = v_1$, which is:
$$[A - \lambda I]v_2 = v_1 \implies [A   - 2 I]v_2 = \begin{bmatrix}
    0       & 1  \\
    0       & 0  
\end{bmatrix}v_2 = \begin{bmatrix} 1  \\ 0 \end{bmatrix} \implies v_2 = \begin{bmatrix} 0  \\ 1 \end{bmatrix}$$
We would write the solution as:
$$X(t) =  e^{\lambda t}(c_1(t v_1 + v_2) + c_2 v_2)$$
So this gives us:
$$X(t) = \begin{bmatrix} x(t)  \\ y(t) \end{bmatrix}  = e^{2 t}\left( c_1(t v_1 + v_2) + c_2 v_2\right) = e^{2t}\left(c_1 \left(t\begin{bmatrix} 1  \\ 0 \end{bmatrix} + \begin{bmatrix} 0  \\ 1 \end{bmatrix}\right) + c_2 \begin{bmatrix} 0  \\ 1 \end{bmatrix}\right)$$
