Is this poset shellable? Let $V$ be a finite dimensional vector space over a finite field $F$. (The case $F = \mathbb{Z}/2\mathbb{Z}$ is the case I most care about.)  Consider the poset of linearly independent subsets of $V$ ordered by inclusion. Is this poset shellable? There is an obvious lexicographic ordering that might be of use.
 A: (Upgrading comments to an answer.)
The collection of linearly independent subsets of a finite-dimensional vector space $V$ over a finite field $F$ ordered by inclusion is naturally a finite simplicial complex: it is evidently the independence complex of a matroid realizable over $F$. 
You might actually want to know if this independence complex is shellable. Asking if the poset of faces is shellable is the same as asking if the barycentric subdivision of this independence complex is shellable.
At any rate: the independence complex of any (finite) matroid is shellable: see Björner, "The homology and shellability of matroids and geometric lattices" (citation below). And moreover,  barycentric subdivision preserves shellability: see Björner, "Shellable and Cohen-Macaulay partially ordered sets".
Hence, whatever your intended question, the answer is: yes, this complex is shellable.
Björner, Anders, Shellable and Cohen-Macaulay partially ordered sets, Trans. Am. Math. Soc. 260, 159-183 (1980). ZBL0441.06002.
Björner, Anders, The homology and shellability of matroids and geometric lattices, Matroid applications, Encycl. Math. Appl. 40, 226-283 (1992). ZBL0772.05027.
