Johannes Hederström: Linear Groups

Abstract.

We present the theory on linear groups. These are defined as subgroups of the general linear group and consists of invertible matrices. Once the theory has been developed we derive several examples and the main topic of study is the special unitary group of \(2 \times 2\) - matrices. We provide a proof that this group is isomorphic to the \(3\)-sphere in \(\mathbf{R}\) and define the equator of this sphere. By constructing a map \(\gamma : \mathrm{SU}_2 \to \{f : E \to E\}\) we show that every matrix of \(\mathrm{SU}_2\) can be represented as an element of the special orthogonal group \(\mathrm{SO}_3\). This representation is interpreted geometrically as a rotation of the \(3\)-sphere. We conclude by considering a class of differentiable homomorphisms. We prove that the image of these homomorphisms define the one parameter groups.