Groups of automorphisms

When one studies the geometrical structure of a vector space, one is often interested in those transformations of the space which leave a certain geometrical quantity invariant. Some examples are as follows.

Let V be a Euclidean space with scalar product 〈·,·〉. Then one can study those (necessarily invertible) endomorphisms of V which are length-preserving (or, equivalently, preserve the given scalar product), i.e., those mappings g: V → V which satisfy ‖gv‖ = ‖v‖ for all v ∈ V or, equivalently, 〈gx, gy〉 = 〈x, y〉 for all x, y ∈ V.

Let F be a finite-dimensional real vector space with a given orientation. Then one can study those automorphisms of V which preserve the orientation; by (8.21) these are exactly the linear mappings g: V → V with det g > 0.

Let V be a finite-dimensional vector space over a field K and let vol be a nontrivial volume function for V. Then one can study those endomorphisms g of V which are volume-preserving in the sense that vol(gv ₁,…,gv_n ) = vol(v ₁,…,v_n ) for all v ₁,…,v_n ∈ V. By (8.12) these are exactly the linear mappings g: V → V with det g = 1.

In special relativity one takes as a simplified model of the universe the “Minkowski space” ℝ⁴ in which an element (x, y, z, t) is considered as an “event” with space coordinates (x, y, z) and time coordinate t. With the velocity of light normalized to be 1, the relativistic “norm” of such an event is q(x, y, Z, t) ≔ t ² ℝ x ² − y ² − z ², and one is interested in those transformations g of ℝ⁴ which preserve the form q in the sense that q(gv) = q(v) for all v = (x, y, z, t) ∈ ℝ⁴.