In this chapter, in preparation for defining the Lie bracket on the Lie algebra of a Lie group, we introduce the adjoint representations of the group \(\mathbf {GL}(n, {\mathbb {R}})\) and of the Lie algebra \({\mathfrak {gl}}(n, {\mathbb {R}})\). The map \(\mathrm {Ad}\colon \mathbf {GL}(n, {\mathbb {R}})\rightarrow \mathbf {GL}({\mathfrak {gl}}(n, {\mathbb {R}}))\) is defined such that AdA is the derivative of the conjugation map \(\mathbf {Ad}_A{\colon } \mathbf {GL}(n, {\mathbb {R}})\rightarrow \mathbf {GL}(n, {\mathbb {R}})\) at the identity. The map ad is the derivative of Ad at the identity, and it turns out that adA(B) = [A, B], the Lie bracket of A and B, and in this case, [A, B] = AB − BA. We also find a formula for the derivative of the matrix exponential exp. This formula has an interesting application to the problem of finding a natural sets of real matrices over which the exponential is injective, which is used in numerical linear algebra.