I've yet to come across a non-geometric definition of two parallel vectors in arbitrary vector (inner-product spaces). Does one exist? is it something like:
$u$ is parallel to $v$ if and only if $\langle u^{\perp}, v \rangle = 0$ for all $u^{\perp} \in \{u^{\perp}\}$?
If so, is it true that two parallel vectors are invariant under (invertible?) linear transformations as they are in $\mathbb{R}^n$? For example, any invertible change of coordinates takes a unit cube in $\mathbb{R}^n$ to an n-parallelipiped. It seems this should be pretty easy to prove in $\mathbb{R}^n$ for any choice of basis, by just using the geometric interpretation of the dot product.
Is there a simple proof of this for invertible linear transformations?