  We establish a new version of Siegel's lemma over a number field $k$,
providing a bound on the maximum of heights of basis vectors of a subspace of
$k^N$, $N \geq 2$. In addition to the small-height property, the basis vectors
we obtain satisfy certain sparsity condition. Further, we produce a nontrivial
bound on the heights of all the possible subspaces generated by subcollections
of these basis vectors. Our bounds are absolute in the sense that they do not
depend on the field of definition. The main novelty of our method is that it
uses only linear algebra and does not rely on the geometry of numbers or the
Dirichlet box principle employed in the previous works on this subject.
