find all the conjugations on $C$ and $C^2$ Let $V$ be a vector space over the complex numbers. A function $J$ from $V$ into $V$ is called a conjugation if $J(a+b)=J(a)+J(b)$, $J(ca)=\bar c J(a)$,
and $J(J(a)=a$ for all scalars $c$ and all $a,b$ in $V$. Here $\bar c$ is the complex conjugate of $c$. 
I do not understand how to apply this definition to the following problem:
If $J$ is a conjugation on a vector space $V$ then the set $W$ of all $a \in V$ such that $J(a)=a$ is a vector space over the real numbers, and for each $a$ in $V$ there exist unique vectors $b$ and $c$ in $W$ such that $a=b+ic$.   

The proof that $W$ is a vector space over the reals is an easy task. It just uses the properties of $J$ and $W$.
If $a=b + ic$ with $b,c\in W$, then
$$
J(a) = J(b) + J(ic) = b - ic.
$$
This implies
$$
a+J(a) = 2b, \quad a-J(a)= 2ic.
$$
Thus setting 
$$
b = \frac12(a+J(a)), \quad c = -\frac i2(a-J(a))
$$
is the desired decomposition. It is easy to check that $b,c$ defined this way are in $W$.
