Transformation to complex spherical basis. According to this Wikipedia page about the complex spherical basis, it is stated that the spherical basis vectors are written in terms of the Cartesian basis vectors the following way:
$$
\left(
\begin{array}{c}
e_{+} \\
e_{-} \\
e_{0}
\end{array}
\right)
=
\left(
\begin{array}{ccc}
-\frac{1}{\sqrt{2}} & -\frac{i}{\sqrt{2}} & 0 \\
+\frac{1}{\sqrt{2}} & -\frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
e_{x} \\
e_{y} \\
e_{z}
\end{array}
\right)
=
\textbf{U}
\left(
\begin{array}{c}
e_{x} \\
e_{y} \\
e_{z}
\end{array}
\right)
$$
However, the "components" of a vector transform by:
$$
\left(
\begin{array}{c}
A_{+} \\
A_{-} \\
A_{0}
\end{array}
\right)
=
\left(
\begin{array}{ccc}
-\frac{1}{\sqrt{2}} & +\frac{i}{\sqrt{2}} & 0 \\
+\frac{1}{\sqrt{2}} & +\frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1
\end{array}
\right)
\left(
\begin{array}{c}
A_{x} \\
A_{y} \\
A_{z}
\end{array}
\right)
=
\textbf{U*}
\left(
\begin{array}{c}
A_{x} \\
A_{y} \\
A_{z}
\end{array}
\right)
$$
I understand that this is what you get by substituting the components of the vectors in the basis definition but, is there a better way to explain why the transformation matrix is conjugated when I go from transforming the basis vectors to transforming the "components" of a vector?

Let $B_c$ denote the Cartesian basis as a column vector. Let $B_s$ be the spherical basis. Then we have $B_s=U B_c.$ If $v$ is a vector, let $v_c$ and $v_s$ be the coordinate representation in each basis. Then
$$
{B_c}^T v_c={B_s}^T v_s=(UB_c)^T v_s={B_c}^T U^T v_s.
$$
But because $B_c$ is a basis this gives $v_c=U^T v_s$ or $v_s=(U^T)^{-1} v_c.$ But because $U$ is unitary,
$$
(U^T)^{-1}=(U^{-1})^T=((U^*)^T)^T=U^*.
$$
