Consider the linear DE $$x'=Ax=\begin{pmatrix}3&-1\\4&-1\end{pmatrix}x,$$ and its general solution $\varphi(t)$. One method would be computing $$\exp(tA)=\begin{pmatrix} (2t+1)e^t & -te^t\\4te^t&(1-2t)e^t\end{pmatrix}$$ and then for any initial value $(x_1,x_2)$ we'd have $\varphi(t)=\exp(tA)(x_1,x_2)$ as a general solution.
Question: I thought that another way to obtain this solution is to just write it as a linear combination of the vectors from the fundamental system. $A$ has eigenvalues $\lambda_1=\lambda_2=1$ so a fundamental system would be $$\left\{v_1=e^t(1,0),v_2=e^t(t,1)\right\}$$but obviously $\varphi(t)\neq x_1 v_1+x_2 v_2$. Why is that? Is my fundamental system wrong?