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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.
If A is a matrix and G is an invertible matrix then det(GAG^{-1}) = det(G)det(G^{-1}) det(A) = det(GG^{-1})det (A) = det(A) Thus , det(A) does not depend on the basis choice, and in fact det(\phi) makes sense for a linear map: \phi: V --> V between finite dim. vector spaces.
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