Algebra and Geometry | Part IA, 2004

The linear map H:R3R3H: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} represents reflection in the plane through the origin with normal n\mathbf{n}, where n=1|\mathbf{n}|=1, and n=(n1,n2,n3)\mathbf{n}=\left(n_{1}, n_{2}, n_{3}\right) referred to the standard basis. The map is given by xx=Mx\mathbf{x} \mapsto \mathbf{x}^{\prime}=\mathbf{M} \mathbf{x}, where M\mathbf{M} is a (3×3)(3 \times 3) matrix.

Show that

Mij=δij2ninjM_{i j}=\delta_{i j}-2 n_{i} n_{j}

Let u\mathbf{u} and v\mathbf{v} be unit vectors such that (u,v,n)(\mathbf{u}, \mathbf{v}, \mathbf{n}) is an orthonormal set. Show that

Mn=n,Mu=u,Mv=v\mathbf{M n}=-\mathbf{n}, \quad \mathbf{M u}=\mathbf{u}, \quad \mathbf{M v}=\mathbf{v}

and find the matrix N\mathbf{N} which gives the mapping relative to the basis (u,v,n)(\mathbf{u}, \mathbf{v}, \mathbf{n}).

Typos? Please submit corrections to this page on GitHub.