A basic fact that I still find a little weird

[sniffnoy]

Suppose we have a matrix representing a linear transformation. Taking its determinant is only a meaningful operation if the domain and the target space are identified and are using the same basis. But the easiest way to compute the determinant is by row reduction (or column reduction), which is only really a meaningful operation if the domain and target bases *don't* need to be the same (and one of them is considered to be fixed).

Is this just how it is ("yup, computation often requires you to do apparently meaningless things") or am I thinking about something wrong? Is there something I'm missing here?

-Harry