Exercise 8, § 7

Let G be a Lie group whose Lie algebra is compact, and which has a finite number of connected components. Show that G has a faithful linear representation on a finite dimensional vector space (write G as the semi-direct product of a compact group K by a vector group N; choose a faithful representation of K on a finite dimensional vector space W and represent G as a subgroup of the affine group of $W\oplus N$).