TITLE: Best way of introducing determinants in a linear algebra course
QUESTION [15 upvotes]: What is the best way of introducing determinants in a linear algebra course? I want to give real life examples of where the determinant is applied. It should have a real impact.

REPLY [4 votes]: This might not be what you want, since "real" applications in linear algebra seem to involve a matrix, and thus a fixed basis.  But the determinant is similarity invariant, meaning that it is basis independent.  This is important because it means that the determinant is a property of the linear transformation regardless of any representation (i.e. the matrix) of that transformation.  This might be a bit advanced, but invariants are important concepts to introduce, and the determinant is a particularly good example of an invariant.