I have recently learned that a solution $\mathbf{x(t)}$ to a $2 \times 2$ system $$\mathbf{x'} = \mathbf{Ax} $$ with $\mathbf{x_0} = c_1\mathbf{u_1} + c_2\mathbf{u_2}$, where $\mathbf{u_1}$ and $\mathbf{u_2}$ are the eigenvectors corresponding to the eigenvalues $\alpha_1$ and $\alpha_2$ of $\mathbf{A}$ ($\alpha_i \in \mathbb{R}$), has the form $$\mathbf{x(t)} = c_1e^{\alpha_1 t}\mathbf{u_1} + c_2e^{\alpha_2 t}\mathbf{u_2}. $$
Then the literature states that with $y_1(t) = c_1e^{\alpha_1 t}$ and $y_2(t) = c_2e^{\alpha_2 t}$, we have $y_2 = \frac{c_2}{|c_1|^{\alpha_2/\alpha_1}} |y_1|^{\alpha_2/\alpha_1}$, and there is no further explanation to why this is true. I was hoping that someone could help to explain this.