The map is defined for , where is a unit vector in and is a constant.
(a) Find the inverse map , when it exists, and determine the values of for which it does.
(b) When is not invertible, find its image and kernel, and explain geometrically why these subspaces are perpendicular.
(c) Let . Find the components of the matrix such that . When is invertible, find the components of the matrix such that .
(d) Now let be as defined in (c) for the case , and let
By analysing a suitable determinant, for all values of find all vectors such that . Explain your results by interpreting and geometrically.