Question

Let L be the line y=mx in R2 and u= [1 m] . Let Qm be the linear transformation that reflects a vector over L.

(a) Show that proj⃗u(⃗v) = A⃗v where
A=(1/ ||u||^2)(uu^T)

(b) Using the fact that the diagonals of a parallelogram bisect each other, show Qm(⃗v) = 2 proj⃗u(⃗v) − ⃗v.

(c) Use Parts (a) and (b) to show that the matrix of Qm is

[(1-m^2)/(1+m^2) 2m/(1+m^2)]
[2m/(1+m^2) (m^2-1)/(1+m^2)]

(d) Show that (Qm ◦ Qm)(⃗v) = ⃗v, for all ⃗v ∈ R .

(e) Determine the matrix of reflection over the line y = x.