Problem 1 Deduce a reflection that reflects a vector $𝐱$ to $\|𝐱\| 𝐞_n$.
Problem 2 A unitary matrix $Q ∈ ℂ^{n × n}$ satisfies $Q^⋆Q = I$ where $Q^⋆ := Q̄^⊤$
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is conjugate transpose. For $𝐱 ∈ ℂ^n$, construct a unitary matrix such that
Q 𝐱 = ± \| 𝐱 \| 𝐞_1
Problem 3 (advanced) The generators of $S₃$ are the permutations that swap two consecutive entries:
\begin{align*}
σ_1 &= \begin{pmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{pmatrix} \\
σ_2 &= \begin{pmatrix} 1 & 2 & 3 \\ 1 & 3 & 2 \end{pmatrix}
\end{align*}
Every $σ \in S₃$ can be written as products of $σ_1$ and $σ_2$. Show that $S₃$ also has a $2 × 2$ representation
that can be deduced from the maps
σ_1 ↦ \begin{bmatrix} 1 \\ & -1 \end{bmatrix} \\
σ_2 ↦ {1 \over 2} \begin{bmatrix} -1 & \sqrt 3 \\ \sqrt 3 & 1 \end{bmatrix}.
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