Relationship between the Identity and Conjugation functions and a real linear T in the Complex Plane

Question

Let $I:\Bbb C\to\Bbb C$ be the identity and $\overline{I}:\Bbb C\to\Bbb C$ be conjugation: $\overline{I}(z)=\overline{z}$. If $T:\Bbb C\to\Bbb C$ is real linear, show there exist unique $w_1,w_2$ in $\Bbb C$ such that $T=w_1I+w_2\overline{I}$.

What I have so far:

Assume $T:\Bbb C\to\Bbb C$ is real linear. Then $T(\lambda z+\gamma w)=\lambda T(z)+\gamma T(w)$ for complex $z,w$ and real $\lambda ,\gamma$.

$I$ is complex linear and $\overline{I}$ is only real linear. Then I tried to model the equation with matrices.

So $w_1(identity matrix)(x,y)+w_2(conjugationmatrix)(x,y)=T$.

Then I multiplied the matrices with the vectors. Then I tried to add the matrices together, and factored out the (x,y). By using the Cauchy-Reimann equations, I came up with $w_1=-w_2$. But if they're additive inverses, they aren't unique.

Help? Thanks.

Answer 1

You are looking for $w_1,w_2\in\Bbb C$ such that $$T(z)=w_1z+w_2\bar z$$ for all $z\in\Bbb C$. We have, in particular, $T(1)=w_1+w_2$ and $T(i)=w_1i-w_2i$, thus \begin{align} w_1&=\frac 12(T(1)-iT(i))& w_2&=\frac 12(T(1)+iT(i)) \end{align} and this proves uniqueness. Since $1$ and $i $ generates $\Bbb C $ over $\Bbb R $, also existence is proved.

Answer 2

Solution using matrices as you want.

The complex number $z = x + yi$ is the columns vector $\pmatrix{x\cr y}$. Obviously: $$ I = \pmatrix{1&0\cr 0&1},\qquad\bar{I} = \pmatrix{1&0\cr 0&-1}. $$ The product by the complex number $u + vi$ is given by the matrix $$ \pmatrix{u&-v\cr v&u}. $$ Then, for any matrix $T = \pmatrix{a&b\cr c&d}$ we want $$ \pmatrix{a&b\cr c&d} = \pmatrix{u&-v\cr v&u}\pmatrix{1&0\cr 0&1} + \pmatrix{s&-t\cr t&s}\pmatrix{1&0\cr 0&-1} = \pmatrix{u + s& t - v\cr v + t&u - s}. $$ You can solve easily the linear system.