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Homework Statement
So, I'm in a 2D Euclidean space V. The two vectors v1 and v2 forms the orthonormal basis set for V.
Consider the transformation T(v1) = a*v1 + b*v2, T(v2) = c*v1 + d*v2
Show, using scalar products (not vector products), that the area scale factor of the transformation T is ad-bc.
Homework Equations
The Attempt at a Solution
The "area scale factor" is simply the determinant in this case. So all I have to do is to show that det([a b;c d]) = ad-bc.
A 2D transformation maps the orthonormal basis set from a unit square to a parallelogram. The area of this parallelogram must then be the area scale factor (since the area of a unit square is 1).
Extending this to 3D and then use the cross product is the simple answer (by definition, the |axb| where a and b are 3D vectors is the area of the parallelogram spanned by a and b)
However, my task is to solve this using the dot product only. Hence, I'm not allowed to extend to 3D. I just cannot see how I'm supposed to do this :-/