Let be the p-dimensional location (or additive) group, and identify each go with the point ce R. Show that the left and right invariant Haar densities for this group are h'

(c) = h'

(c) = 1.

The following three problems deal with groups, , of matrix transformations of RP. We will identify a transformation with the relevant matrix, so that ge is a

(pxp) matrix (with (i,j) element gi), and the transformation is simply x→gx. The composition of two transformations (or the group multiplication) corresponds simplyto matrix multiplication. Thus to find Haar densities, we must be concerned with the transformations of & given by g→gg and g→gg. To calculate the Jacobians of these transformations, write the matrices g, gº, gog, and gg asas vectors, by stringing the rows of the matrices end to end. Thus a (pxp) matrix will be treated as a vector in RP. Coordinates which are always zero can be ignored, however, possibly reducing the dimension of the vector. The densities calculated for the transformations are then densities with respect to Lebesgue measure on the Euclidean space spannedby the reduced vectors.