Invariance of vectors due to changes in coordinate systems

[etothey]

Homework Statement

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?

Homework Equations

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

The Attempt at a Solution

If ds=ds1

Is this correct?

 

[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.

 

[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?

 

[Chestermiller]

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?

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.

 

[Zamze]

Thank you

 

[kevinferreira]

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.

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.