# difference between vector space and fundamental matrix - ODE

The solution(s) of a system of first order differential equations seem to be contained in a vector space and as well in a fundamental matrix in the form of columns.

Could someone please explain a litlle more the difference between both. And/or correct my statement if wrong?

The solutions of an equation $x'=A(t)x$ for some $n\times n$ matrices $A(t)$ varying continuously with $t$ are of the form $x(t)=X(t)c$, where $X(t)$ is an $n\times n$ matrix whose columns form a basis of the (vector) space of solutions of the equation and $c\in\mathbb R^n$ is arbitrary. Note that $X(t)$ is never unique, although the vector space of solutions is uniquely determined.
Added: In general the basis need not be orthogonal. What is above means that you have $n$ functions $x_1(t),\ldots,x_n(t)$ (with values in $\mathbb R^n$) such that any solution can be written in the form $$x(t)=c_1x_1(t)+\cdots+c_nx_n(t)=\begin{pmatrix} x_1(t) & \cdots x_n(t)\end{pmatrix} \begin{pmatrix} c_1\\ \vdots\\ c_n\end{pmatrix}.$$ and so you can take $$X(t)=\begin{pmatrix} x_1(t) & \cdots x_n(t)\end{pmatrix},$$ where each $x_i(t)$ is a column.
• The answer is "yes" (always, for any system). On the other hand, the answer to your second question is "no" (also never). The example with $2X(t)$ shows precisely this. On the other hand, the product $X(t)X(t_0)^{-1}$ is independent of the particular $X(t)$ chosen. – John B Jan 22 '16 at 21:49