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The determinant of a set of n vectors in R^n is defined to be the antisymmetric function det: R^n x R^n x... x R^n -> R linear in each argument such that
det( e_1, ...., e_n) = 1.
This is well-defined because the signature of a permutation is well-defined.
By antisymmetry and multilinearity we see that the (signed) volume of the parallelipiped generated by vectors v_1, v_2, ..., v_n is just det(v_1, ..., v_n).
A set of vectors v_1, ..., v_n generates the whole space if and only if the parallellipiped they span has nonzero volume, i.e. det(v_1, ..., v_n) =/=0.
A linear transformation f:R^n -> R^n is invertible if and only if the images f(e_i) span the whole R^n if and only if det(f(e_i))!=0.
It's all so simple and so illuminating. I can't understand how anyone can go through linear algebra without understanding the geometric aspect of it (major part of which is given by the determinant).
