Take the tour ×  Given is a second-order tensor $T$, and three
arbitrary vectors, $u$, $v$ and $w$, defined in Euclidean point
space $\mathcal{E}$.  Prove that the determinant of the tensor
$T$  $\det T=\frac{Tu.(Tv \times Tw)}{u.(v \times w)}$  is
independent of $u$, $v$ and $w$, as long as they form a base in
the 3-dimensional vector space $\mathcal{V}$ defined on
$\mathcal{E}$ (that is, vectors $u$, $v$ and $w$ are linearly
independent).  This is basically proving that the determinant of
a tensor is invariant of the particular basis it happens to be
resolved in.  I've tried to work through this using index
notation, but it got pretty cumbersome quickly, so I thought
there might be another way to do it. I'm not looking for the
full treatment here, but rather as a general advice how to
tackle this. I think I can manage the details myself.
share|improve this question add comment  2 Answers  Since you
use the cross and dot products in your question, you are working
over an inner product space, therefore $\mat