Multilinear Maps of V^n into Reals and Effect of Linear Transforma

[WWGD]

Hi, All:

Is there a "nice" , non-messy way of showing this:

Let M(v_1,v_2,..,v_n) → R⁺, where R is the Reals, be a multilinear map,

where v_i are vectors in a finite-dimensional vector space V.

Now, let L: Vⁿ → Vⁿ be a linear map with Det(L)>0 .

How do we show that M(L(v_1,v_2,..,v_n))( is also strictly-positive? I think it

has to see with the fact that the map L preserves the orientation of Vⁿ,

but I don't see how to make this more rigorous. Any ideas?

 

[fresh_42]

I would write $M,L$ as tensors and search whether there are some nice formulas which connect tensors and determinantes, or otherwise go the hard way by coordinates.