Calculate the spectral norm 
Consider the four vectors $v_1, v_2, u_1, u_2 \in \mathbb{C}^2$ with
$$v_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \qquad v_2 = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \qquad u_1 = \begin{pmatrix} 1 \\ i \end{pmatrix}, \qquad u_2 = \begin{pmatrix} 2+i \\ 1 \end{pmatrix}$$
Find the spectral norm of $\phi: \mathbb{C}^2 \to \mathbb{C}^2$ that maps the vector $v_i$ to the vector $u_i$.

Before people downvote: I know that the spectral norm is given as the square root of the max eigenvalue of the matrix $B$ with $B=A^{H}A$ where $A$ is the matrix representation of $\phi$. However, I struggle with the format of the task given as there is no matrix given and the vectors are not orthonormal.
 A: Hint:
You have to find the images of the canonical basis $(e_1,e_2)$ knowing that
$$\begin{cases}\phi(v_1)=\phi(e_1)+\phi(e_2)=\begin{bmatrix}1\\i\end{bmatrix}\\[1ex]
\phi(v_2)=2\phi(e_1)+\phi(e_2)=\begin{bmatrix}2+i\\1\end{bmatrix}\end{cases}.$$
A simple elimination shows
\begin{align*}
\phi(e_1)&=\phi(v_2)-\phi(v_1)=\begin{bmatrix}1+i\\1-i\end{bmatrix},\\[1ex]
\phi(e_2)&=2\phi(v_1)-\phi(v_2)=\begin{bmatrix}-i\\2i-1\end{bmatrix}.
\end{align*}
A: We have two linear systems in $\mathrm A \in \mathbb C^{2 \times 2}$
$$\mathrm A \mathrm v_1 = \mathrm u_1 \qquad \qquad \qquad \mathrm A \mathrm v_2 = \mathrm u_2$$
Vectorizing, we obtain the following linear system of $4$ equations in $4$ unknowns
$$\begin{bmatrix} \mathrm v_1^T \otimes \mathrm I_2\\ \mathrm v_2^T \otimes \mathrm I_2\end{bmatrix} \mbox{vec} (\mathrm A) = \begin{bmatrix} \mathrm u_1 \\ \mathrm u_2\end{bmatrix}$$
Once we have $\mbox{vec} (\mathrm A)$, we un-vectorize it to obtain $\mathrm A$. Lastly, we compute the spectral norm $\|\mathrm A\|_2$.
