Define the groups $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$.

Show that $G=\mathrm{SU}(2)$ acts on the vector space of $2 \times 2$ complex matrices of the form

$$V=\left\{A=\left(\begin{array}{cc} a & b \\ c & -a \end{array}\right) \in \mathrm{M}_{2}(\mathbb{C}): A+\overline{A^{t}}=0\right\}$$

by conjugation. Denote the corresponding representation of $\mathrm{SU}(2)$ on $V$ by $\rho$.

Prove the following assertions about this action:

(i) The subspace $V$ is isomorphic to $\mathbb{R}^{3}$.

(ii) The pairing $(A, B) \mapsto-\operatorname{tr}(A B)$ defines a positive definite non-degenerate $\mathrm{SU}(2)$ invariant bilinear form.

(iii) The representation $\rho$ maps $G$ into $\mathrm{SO}(3)$. [You may assume that for any compact group $H$, and any $n \in \mathbb{N}$, there is a continuous group homomorphism $H \rightarrow \mathrm{O}(n)$ if and only if $H$ has an $n$-dimensional representation over $\mathbb{R}$.]

Write down an orthonormal basis for $V$ and use it to show that $\rho$ is surjective with kernel $\{\pm I\}$.

Use the isomorphism $\mathrm{SO}(3) \cong G /\{\pm I\}$ to write down a list of irreducible representations of $\mathrm{SO}(3)$ in terms of irreducibles for $\mathrm{SU}(2)$. [Detailed explanations are not required.]