To find the inverse transformation S-1 of S and its determinant, we invert the matrix representing S and calculate the determinant of S to take its reciprocal. The area of the parallelogram spanned by S-1(a) and S-1(b) is found using the cross product of these vectors. The determinants of S and S-1 are reciprocals of each other, which is generally true for any invertible linear transformation.
The student asks to find the inverse of the transformation S represented by x → (BA)x along with its determinant, graph the parallelogram spanned by the inverse of two vectors a and b, and enquire about the relationship between the determinants of S and S-1.
The inverse of a linear transformation S is the transformation that, when composed with S, results in the identity transformation. This can be represented in matrix terms as S-1, where SS-1 = I. To find S-1, we would take the inverse of the matrix representing S, which is assumed to be the product of two matrices B and A. To determine the determinant of S-1, we would calculate the determinant of the matrix representation of S and take the reciprocal.
For the parallelogram, the vectors S-1(a) and S-1(b) define its sides. To find its area, we can take the cross product of these vectors (or, in this case, the determinant of the matrix formed by placing these vectors as columns or rows) and take the absolute value. The relationship between the determinants of S and S-1 is such that the product of these determinants is 1, which is true in general for any invertible linear transformation.