Linear independence of solutions of time-dependent linear dynamical system

Question

I have a time-dependent linear system of differential equations $$\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t)$$. In the time-independent case, $\det{\mathbf{A}}\ne 0$ ensures there are $n$ linearly-independent solutions, how do these conditions change in the time-dependent case?

Answer 1

The Picard–Lindelöf theorem ensures that there are $n$ linearly independent solution where $n$ is the dimension of the real space we're working in (I assume that this is the meaning of $n$ in your question).

The proof is quite simple. Assume that $x_i$ is the solution of the IVP problem

$$\dot{\mathbf{x}}(t) = \mathbf{A}(t) \mathbf{x}(t)$$

where all the coordinates of $x_i(0)$ are vanishing except the $i$-th one which is equal to $1$. Then those solutions are linearly independent.

And by the way, this is true whatever the value of $\det A(t)$. You don't have to suppose that the determinant is not vanishing.