Let be a linear transformation and be a rectangle.

Prove:

(1) Let be the standard basis vectors of (i.e. the columns of the identity matrix). A permutation matrix is a matrix whose columns are , , where is a permutation of the set . If , then .

(2) let be an matrix where has exactly one non-zero entry with . If , show that .

(3) Recall that a matrix is elementary if is a permutation matrix as in (1), or as in (2), or is diagonal with all but one diagonal entry equal to . Deduce that if and is an elementary matrix, then for any Jordan domain , .

(4) Recall from linear algebra (row reduction), that any invertible matrix is a product of elementary matrices. Prove that for any Jordan domain , , where is invertible.

(5) Is (4) true if we do not assume is invertible?

(6) Prove: If is an affine transformation and is a Jordan domain, then where is the derivative of at some point .

((1) and (2)are easy but I have little ideas about the rest. What's the volume of a Jordan domain and what's the relationship between and ? Why for rectangle but for Jordan domain, ?) Thank you.