G(t) is nxn matrix dependent on t. show solutions of x' = G(t)x form an n-dim subspace of the space of functions C^1(R^+,R^n) .
Not sure how to do this.
Ofcourse, $\displaystyle G$ has to be continuous, or else $\displaystyle x$ might not be $\displaystyle C^1$.
Consider the n (unique) solutions to the n problems $\displaystyle \{x_i'=Gx_i, \ x(0)=e_i\}_{1\leq i\leq n}$, where $\displaystyle \{e_i\}$ are the standard basis vectors.