Consider a $C^k$ function $t \mapsto A(t)$ defined on the interval $(a, b)$ and taking its values in the set of real $n \times n$ matrices. If $\displaystyle\sup_{t \in (a, b)}|||A(t)||| < +\infty$ where $|||\cdot|||$ is the operator norm, then the solutions of the vector ordinary differential equation $$ \dot X(t) = A(t) X(t) $$ form a $n$-dimensional subset of $C^{k+1}(a, b)$.
Is this also true if we consider this differential equation in the sense of distributions?