 Solution

A Givens rotation acts as the identity matrix outside two selected coordinates $i,j$, and has the following block on those coordinates:The convention with the opposite signs of the sines is equivalent after replacing $\theta$ by $-\theta$. A Householder reflection with nonzero normal vector $w\in {\mathbb{R}}^n$ isIt reverses the component parallel to $w$ and fixes the perpendicular hyperplane. Both are orthogonal matrices, so their transposes are their inverses and they preserve the Euclidean norm.