Recall as in lecture our axioms for the volume of a parallel

Recall as in lecture our axioms for the volume of a parallelepiped: Given v_1,...,v_n vectors in R^n we have vol (v_1,...,v_n) epsilon R_Greaterthanorequalto 0 such that (with c epsilon R): vol(e_1,...,e_n) = 1 vol(v_1,...,ev_i,...,v_n) = |c| vol (v_1,...,v_i,...,v_n) (here v_i + cv_j is in the i-th spot, with i notequalto j) vol(v_1,...v_i,...,v_j,...,v_n) = vol(v_1,...,v_j,...,v_i,...,v_n) (here v_i and v_j are in the i-th and j-the spots on the LHS, and swap on the RHS) Using these axioms, prove that |det(v_1,...,v_n)| = vol(v_1,...,v_n).

Solution

we have

|det(e1,......en)|=1

|det(v1,....,cvi,....,vn)|=|c||det(v1,....,vi,....vn)|

|det(v1,....vi+c vj,...vn)|=|det(v1,....vi,....vn)|

|det(v1,....vi,.....vj,....vn)|=|det(v1,....vj,.....vi,....vn)|

all these are the properties of determinent

|det(v1,....vn)|=vol(v1,...vn)