Recall the phase portrait of the linear system $\dot{x}=Ax$ with $A = \begin{pmatrix} -1 & -3 \\ 0 & 2 \end{pmatrix} $ Describe $\phi_{t}(N_{\epsilon}(x_0))$ for $x_0=(-3,0)$ , $\epsilon=.2$ where $\phi$ denotes the flow map $\phi_{t}=e^{tA}$
Solution so far
The eigenvalues and eigenvectors $\lambda_{1}=2$, $\lambda_{2}=-1$, $v_1=\begin{pmatrix} -1 & 1 \end{pmatrix}$, and $v_2=\begin{pmatrix} -1 & 0 \end{pmatrix}$ $\implies \phi_{t}(N_\epsilon(x_0))= \begin{pmatrix} e^{2t} & 0 \\ 0 & e^{-t} \end{pmatrix}x$ I am not understanding how to describe the flow
The flow is not what you wrote but instead $$ \phi_t=\begin{pmatrix} -1 & -1\\0 & 1\end{pmatrix} \begin{pmatrix} e^{-t} & 0\\0 & e^{2t}\end{pmatrix} \begin{pmatrix} -1 & -1\\0 & 1\end{pmatrix}^{-1}=\begin{pmatrix} e^{-t} & e^{-t}-e^{2t}\\0 & e^{2t}\end{pmatrix}. $$ The rest will come from your computations.