The map is defined for , where is a unit vector in and is a real constant.
(i) Find the values of for which the inverse map exists, as well as the inverse map itself in these cases.
(ii) When is not invertible, find its image and kernel. What is the value of the rank and the value of the nullity of ?
(iii) Let . Find the components of the matrix such that . When is invertible, find the components of the matrix such that .