1. The problem statement, all variables and given/known data A linear vector field on R^n, defined by a matrix A, is a Killing field if and only if A is antisymmetric. 2. Relevant equations 3. The attempt at a solution I took R^2. Let A be the following matrix: 0 -10 10 0 A is antisymmetric. However, how in the heck is A a Killing field? For example, let v = (1, 10), then Av = (-100, 10) u = (2, 3), then Au = (-30, 20) and <u, v> = 32 while <Au, Av> = 3200, so the dot product after we apply A is a multiple of the original dot product by the determinant of A (in this case it's 100). What exactly am I misreading? The only restriction on A is that it is antisymmetric. A Killing field is a vector field that preserves the metric, i.e. in particular it will preserve the dot products of vectors.