A problem in my professor's guide is driving me nuts, I don't even know where to start, this is the problem:
Is $ \{(z,u) \in \mathbb{C}^2 / z - \overline{z} + u = 0\} $ a sub-space of $ (\mathbb{C}^2,+,\mathbb{R},.)? $ If it is, find its base and dimension.
I have solved similar problems to this one, the only difference is that the other ones didn't have complex numbers, also, what does $\overline{z}$ means in this problem?
Edit: I'm not asking you to solve the problem, just point me out in the right direction!
Recall that every complex number can be written in the form $z=a+bi$, where $i$ is the imaginary unit.
Then with $z$ as above, $\overline{z}$ is defined to be $\overline{z}=a-bi$.
Thus, $z-\overline{z}+u=0$ means that $a+bi-(a-bi)+u=2bi+u=0$, and thus $u=-2bi$. So the elements of the set you are looking at are of the form $(a+bi,-2bi)$.
To try and demystify the use of complex variables, another way you can think of the complex number $a+bi$ is as the ordered pair $(a,b)$. In this case, the pair of complex numbers $((a,b),(c,d))$ can just be identified as the ordered quadruple $(a,b,c,d)$. Thus, the set of points you're looking at are the quadruples $(a,b,0,-2b)$, where addition and scalar multiplication are defined component-wise.