Invariance of vectors due to changes in coordinate systems

1. Nov 25, 2012

etothey

1. The problem statement, all variables and given/known data

How do I know that vector is invariant to changes of coordinate systems if i only have the components of the vector and not the basis vectors?

2. Relevant equations
let the vector in reference frame 1 be ds and the same vector in the reference frame 2 be ds1

3. The attempt at a solution

If ds=ds1

Is this correct?

2. Nov 25, 2012

haruspex

Not at all sure I understand the question.
A vector is usually thought of as existing independently of any co-ordinate system within which to represent it. So by definition the vector itself is invariant, but its representation is not.

3. Nov 26, 2012

Zamze

Hi!

Id like to point out that I am the guy that made the post, accidentally logged in with the wrong account. So the exact question is

How do we know that a vector is invariant to changes of coordinate system if we only have the components of the vector and not the basis vectors?

4. Nov 26, 2012

Staff: Mentor

If you have the components in two coordinate systems, and the components are related by the proper transformation law, then the components are those of an invariant vector.

5. Dec 20, 2012

Zamze

Thank you

6. Dec 25, 2012

kevinferreira

Exactly. This means that if $X^{\mu},~X'^{\mu}$ are the components of your vector in two different basis $(x^{\mu}),~(x'^{\mu})$ you should have
$$X'^{\mu}=\frac{\partial x'^{\mu}}{\partial x^{\nu}}X^{\nu}$$
In this way $X = X^{\mu}E_{\mu}$ represented on a basis $(E_{\mu})$ is an invariant object, as it should.