I am trying to solve this exercise from Linear Algebra Done Wrong:
- If I “identify” a complex number $x + iy$ with a vector $(x, y)^T$ then does multiplying by a complex scalar $\alpha = a + ib$ result in the vector $(\alpha x, \alpha y)^T$ or in the vector $(ax - by, ay + bx)^T$?
- In (b), I don’t understand what it means to multiply by $\alpha = a + ib$ if we treat $\mathbb{C}$ as a real vector space, if in a real vector space the scalars are real numbers and not complex numbers.
If someone could offer some insights on these questions I would appreciate it!