Matrix form for conjugate linear transformation I knew that every linear transformation has matrix representation. I wonder whether I could have a matrix representation for conjugate linear transformation. By conjugate linear transformation, I mean under scalar multiplication instead of $C(af)=aC(f)$, I would have $C(af)= \overline{a} C(f)$, where $a$ is a constant complex number, and $C$ is the transformation.
Thank you so much!
 A: Given a linear-conjugate function $C: V \longrightarrow V$, $V$ a complex vector space, we define the conjugate vector space $\overline{V}$, by being the same set $V$, with an altered multiplication by scalar $\ast$, defined by $$\lambda \ast x := \overline{\lambda} \cdot x $$
where $\cdot$ is $V$'s usual multiplication. See the notations, $V = (V, +, \cdot)$, and $\overline{V} = (V, +, \ast)$. I reforce, the set of vectors is the same. It should be checked that $\overline{V}$ is indeed a vector space, although it's easy to see. Some facts are:
i) a set $\mathcal{B}$ is a basis for $V$ iff it is a basis for $\overline{V}$;
ii) from i) follows that $\dim V = \dim \overline{V}$;
iii) $W$ is a subspace of $V$ iff it is a subspace of $\overline{V}$;
iv) Given a basis $\mathcal{B}$, $x \in V$ has coordinates $(x_1, \cdots, x_n)_\mathcal{B}$ iff considering $x \in \overline{V}$, we have $x = (\overline{x_1}, \cdots, \overline{x_n})_\mathcal{B}$;
v) $C: V \rightarrow V$ is linear conjugate iff $C: \overline{V} \rightarrow{V}$ is linear iff $C: V \rightarrow \overline{V}$ is linear;
vi) $\overline{\overline{V}} = V $
Using this construction, you can use everything you already know.
