  Consider a real matrix $\Theta$ consisting of rows
$(\theta_{i,1},\ldots,\theta_{i,n})$, for $1\leq i\leq m$. The problem of
making the system linear forms
$x_{1}\theta_{i,1}+\cdots+x_{n}\theta_{i,n}-y_{i}$ for integers $x_{j},y_{i}$
small naturally induces an ordinary and a uniform exponent of approximation,
denoted by $w(\Theta)$ and $\widehat{w}(\Theta)$ respectively. For $m=1$, a
sharp lower bound for the ratio $w(\Theta)/\widehat{w}(\Theta)$ was recently
established by Marnat and Moshchevitin. We give a short, new proof of this
result upon a hypothesis on the best approximation integer vectors associated
to $\Theta$. Our conditional result extends to general $m>1$ (but may not be
optimal in this case). Moreover, our hypothesis is always satisfied in
particular for $m=1, n=2$ and thereby unconditionally confirms a previous
observation of Jarn\'ik. We formulate our results in the more general context
of approximation of subspaces of Euclidean spaces by lattices. We further
establish criteria upon which a given number $\ell$ of consecutive best
approximation vectors are linearly independent. Our method is based on Siegel's
Lemma.
