Linear transformation over C and R respectively

Question

I am new to linear algebra, and was asked to determine if the following is a linear transformation.

$T(z_1,z_2)=z_1+\overline z_2, T_3> C^2 \to C$

a) when we look at $C, C^2$ as linear spaces over C.

b) when we look at $C, C^2$ as linear spaces over R.

I have an answer, but I fear it is lacking in many ways:

a)

I tried to give a counterexample:

$z_1=1+1i$

$z_2=2+2i$

$λ$ is a scalar 3+3i $\in C$

Therefore, $λz_1=6i$, $λz_2=12i$

So:

$T(λz_1,λz_2)=(6i,12i)=6i-12i=-6i$

$λT(1+1i,2+2i)=λ(1+1i+2-2i)=(3+3i)(3-i)=12+6i$

$-6i \neq 12+6i$

So the answer is no...

b) using scalars α,β $\in R$

$T(α(z_1+z'_1),β(z_2+z'_2))=α(z_1+z'_1)+ β(\overline z_2+\overline z'_2)$ which is not equal to $αT(z_1+z'_1)+βT(z_2+z'_2)$, so the answer is no.

Could anyone help out?

Thank you!

Answer 1

(a) Looks good.

(b) This is incorrect. $T$ is linear in this case. Remember, the criteria for linearity is that $aT(z_1,z_2)=T(a(z_1,z_2))$ and $T((z_1,z_2)+(z'_1,z'_2))=T(z_1,z_2)+T(z'_1,z'_2)$.