Can you multiply imaginary unit i and a basis vector together?

Question

A while ago I stumbled apon this multiplication:$$i \begin{bmatrix} 0\\ 1 \end{bmatrix} $$ I then remembered that the matrix: $$ J= \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} $$ Holds the property that $J^2=-I$ and so I substituted it as $i$ and ended up with the result that $$ i \begin{bmatrix} 0 \\1 \end{bmatrix} = \begin{bmatrix} 1 \\0 \end{bmatrix} $$ I was suprised when after that I got a correct answer to the problem I was solving. Now I am wondering, if you are allowed to do this kind of substitution and why/why not? Would it be correct to associate this answer to the rotational nature of both $i$ and $J$?

Answer 1

Multiplying by $i$ is a 90-degree counterclockwise rotation. Multiplying by $J$ is a 90-degree clockwise rotation. (So, you can substitute $J$ as $-i$, not $i$.) Both $i^2$ and $J^2$ give $-I$, because rotating 180 degrees in either direction give the same result, namely, flip the vector in the opposite direction.

Perhaps you got the correct answer to your problem because you only had to apply the rotation twice; in this case, substituting $J$ with $i$ or $-i$ gives the same thing.