Determinant = volume using rows.

[Damned charming :)]

Is one way of looking of the determinant is its the area of the parallelogram formed by the vectors in 2 dimensions, the volume of the parallelpided in 3 dimensions etc. The sign of the determinant tells you something about the relative position of the vectors. This would make the diagonalisation process the transform that turns the parallelogram into a square, and the parallelpiped into a cube.

[matt grime]

Is that a question?

Let L denote the n'th exterior power of R^n. Any linear map induces a linear transformation on L. This is just a number. That number is the determinant.