(a) Consider the matrix
Find the kernel of for each real value of the constant . Hence find how many solutions there are to
depending on the value of . [There is no need to find expressions for the solution(s).]
(b) Consider the reflection map defined as
where is a unit vector normal to the plane of reflection.
(i) Find the matrix which corresponds to the map in terms of the components of .
(ii) Prove that a reflection in a plane with unit normal followed by a reflection in a plane with unit normal vector (both containing the origin) is equivalent to a rotation along the line of intersection of the planes with an angle twice that between the planes.
[Hint: Choose your coordinate axes carefully.]
(iii) Briefly explain why a rotation followed by a reflection or vice-versa can never be equivalent to another rotation.
Part IA, 2021 List of Questions