I am trying to solve for $\mathbf x$ (a complex vector) in the equation:
$$i\mathbf{x} - 3\mathbf{v} = \mathbf{u}$$
where $\mathbf{u} = \langle3-4i, 2+i, -6i\rangle$ and $\mathbf{v} = \langle 1+i, 2-i, 4\rangle$.
I did this bit of algebra: \begin{align} i\mathbf{x} - 3\mathbf{v} = \mathbf{u} &\implies i\mathbf{x} = \mathbf{u} + 3\mathbf{v} \\ &\implies \mathbf{x} = -i(\mathbf{u}+3\mathbf{v}) \\ &\implies \mathbf{x} = -i\mathbf{u} -3i\mathbf{v} \end{align}
Which gave me $\mathbf{x} = \langle-1-6i, -2-8i, -6-12i\rangle$.
However the solution guide to my textbook says the answer is:
$$\mathbf{x} = \langle 7 − 6i, −4 − 8i, 6 − 12i\rangle$$
It seems the complex components of my answer are the same as those in the solution, and that there's some sort of multiplication by $-1$ happening to the real parts of $\mathbf{u}$ or $\mathbf{v}$. So I'm wondering if there's some kind of subtlety in the properties of complex numbers that I'm missing. Or maybe my algebra is wrong, or maybe I'm just hopelessly confused.
Any help would be much appreciated, thank you in advance.
Turns out the textbook was wrong and I got the right answer. thank you to ZeroXLR who reminded me I could just plug x into the original equation to check if it holds. thank you to Brian who reminded me since the complex vector space is a vector space, all 8 vector space axioms must hold, which includes k(u+v) = ku + kv