can someone explain me what a volume scale factor is and if I have a liner Transformation
T(v1) = av1 + bv2
T(v2) = cv1 + dv2
how can I find the volume scale factor of it?
You mean area factor? (Volume factors are for 3D...)
Think of it this way: if you have an enlargement matrix, say scale factor 2, then your matrix is .
So, if we define a unit square (1x1 length) using four points, say and .
Then if we put all those points into the matrix, the transformed points are and .
These points define a matrix with an area of 4 square units, so the original unit square got enlarged by an area factor of 4. That is to say, if you used points to define a polygon with area n square units, then the transformed shape will have an area of 4n square units.
For any generic transformation matrix, the area factor is equal to the determinant.
i.e: For any matrix , the area factor is (for a 2x2 matrix).