Did I fill the table about vectors correctly?

Question

I filled the table below. Did I do as it should be? Thanks

I made a cross because I thought there is no a corresponding vector as the dimension had called.

Answer 1

No. (sorry, I was going to post this as a comment, but they are too many things to say)

But let's be a bit more precise.

1. Your entries in the first columns are not consistent, e.g. $(1 2)$ is a tuple, while the other ones are (I suppose) scalars?
2. Assuming $(1 2)$ is a two dimensional vector, then it shouldn't have s representation in $\mathbb{R}^3$. I'm guessing the number $1$ and $2$ are coefficients in a certain (here canonical?) basis, it could be the vectors associated with the traditionally-used $x$ and $z$ or $y$ and $z$ in which case your representation is not correct. But in any case, we cannot tell, as it could be in an even more complex representation. Say, $1$ and $2$ are the coefficients in a basis of the plane defined by $x+y+z = -1$, then none of the results you gave (in this row) are correct.
3. What are $i$, $j$, $k$? Are these the quaternionic complex units? Then the representations in $\mathbb{R}^3$ are not correct. Neither are they in $\mathbb{R}^2$.
4. Are they vectors (and not complex units)? Then I'm $2i$ should have a notation in $\mathbb{R}^1$ ...

These are just the first few things I can think of. The question is not clear and fails to give the context, the definitions of things, and the goal.