Vector Components and Uniqueness

[DJBruce]

So here is a question I am struggling with:

 

"There is only one unique coordinate system in which vector components can be added. True or False?"

 

I believe this is false, as I think you could create numerous different coordinate systems, and all you would have to do is make sure you watch what signs you assign to each vector. However, I am really not very comfortable with this.

[timo]

I'm not sure if I understand the question. But I'd assume the CS in which addition holds and then either rotate or (even easier) rescale it. Component addition should work in this new system, too. I am assuming that you speak of vector spaces, though.

[ajb]

All you really have to do is think about linear transformations. You can prove that the sum of two vectors (their components) is still a vector, i.e. it transforms in the correct way.

 

If we have [math]X^{\alpha}[/math] as a component in some coordinate system then in some other system we have [math]\overline{X}^{\alpha} = X^{\beta}T_{\beta}^{\:\: \alpha}[/math]. Now see what happens to a sum.