I have recently started taking a Linear Algebra course and we have been given a question beyond what we have studied so far. I was hoping I could find some guidance here.
We are told to find out whether $ f:\ \mathbb{C} \to \mathbb{C} $ is a linear transformation or not. It is defined as: $$\forall z=x+iy,\; ((x,y)\in \mathbb R^2),\; f(z) = f(x + i y) = i y $$
Then we have to find the Kernel of this transformation.
Hint: if $x=a+ib$ and $y=c+id$, given $x+y=(a+c)+i(b+d)$, what is $f(x+y)$?
Same if $\lambda \in \mathbb{R}$ or $ \mathbb{C}$, what is $\lambda x$ and $f(\lambda x)$? Note that the domain of $\lambda$ is important. Do the calculation and you will see that $f$ can be linear on $\mathbb{C}$ seen as a vector space on $\mathbb R$ and maybe not linear on $\mathbb C$ seen as a vector space on $\mathbb C$.
PS: if you represent $\mathbb C$ as the real plane $\mathbb R^2$ (so as a $\mathbb R$-vector space), what is the geometric interpretation of $f$ ? (Hint: it has something to do with the projection on a line).
Also, note that $f(z) = i \cdot \Im(z)$, maybe you've learnt properties of the "Imaginary part" function.
Hint for the second question: for the kernel, just write $z=x+iy\in \ker f \Leftrightarrow f(z)=iy = 0$ and conclude. (Re-hint: the kernel is a "line" in $\mathbb R^2$).