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.

 

[M98Ranger]

Yes, that is correct. The determinant is a measure of the size and orientation of a geometric object formed by vectors. In 2 dimensions, it represents the area of the parallelogram formed by the vectors. In 3 dimensions, it represents the volume of the parallelpiped. The sign of the determinant tells us about the relative orientation of the vectors, whether they are parallel, perpendicular, or at an angle to each other. In the process of diagonalization, the transformation essentially turns the parallelogram into a square and the parallelpiped into a cube, making the calculations easier to understand and visualize.