michael_nielsen

The determinant of a matrix is how much the matrix expands or contracts space. More precisely: it's the volume of the parallelepiped spanned by the columns of the matrix. (Ditto the rows.) This, btw, is why det 0 => non-invertible: one or more spatial dimensions has collapsedhttps://twitter.com/ZachWeiner/status/1073671743846395905 …

That's one way. There are many other useful equivalent ways of thinking of the det: (1) As the product of the eigenvalues (for normal matrices); (2) as the product of the singular values (up to sign) for any matrix; (3) as the value of the wedge product of the columns c1^c2^ ...

(4) as a matrix function which (a) rescales when you rescale a column, (b) isn't changed by addition of one column to another, and (c) changes sign when columns are swapped. Fun to think through why all these are equivalent! Many other ways of thinking of the determinant, too.

Some considerable fraction of mathematics seems to be keeping multiple ways of thinking about an object in your head, and moving back and forth between them.

Learning new math concepts often comes down to identifying the "primitive" object and then figuring out how the new concept extends this primitive (credit to @geomblog for impressing this idea). Having multiple interpretations makes it much easier to see extensions