# Transformation of four-vectors and basis vectors

I was taught that vectors are one rank tensors, so under diffeomorphism they do not change (their components do change but not them as tensors) they are contravariant, i.e. $$\begin{equation} A = A^{\mu} e_{\mu} = \frac{\partial x^{\mu}}{\partial x^{\nu}} A^{\nu} \frac{\partial x^{\rho}}{\partial x^{\mu}} e_{\rho} = A^{\nu} e_{\nu} \end{equation}$$ But, why do the basis vectors transform but the $$A$$ vector doesn't? Are they different types of vectors?

• The tensor $A$ doesn't change - only the components of the tensor - $A^{\mu}$ - change when you change coordinate systems. – Cinaed Simson Aug 13 at 23:07
• This is linear algebra. Do you know the difference between a vector and a basis of a vector space? – DanielC Aug 13 at 23:19
• DanielC: A basis of a vector space is a set of linearly independent vectors that generate the whole space, in this sense I don't see a wide difference. – Geovanny Alexander Rave Franco Aug 14 at 3:13
• Cinaed: Yes, that's my question, but I asked to my Cosmology Professor and he told me that we have to think about the vectors as differential operators and, in that sense, $A$ and $e_{\mu}$ are different differential operators so the do not have to transform in the same way. That seems a good answer for me but I'd like to read more answers. – Geovanny Alexander Rave Franco Aug 14 at 3:18

## 1 Answer

In terms of a differential operator - also known as the natural coordinate system, I would write $$A=A^{i}\frac{\partial}{\partial u^{i}}$$.

Then transforming the$$\frac{\partial}{\partial u^{i}}$$ basis to the $$\frac{\partial}{\partial w^{j}}$$ basis, for instance, I would use the rule $$\frac{\partial}{\partial u^{i}}=\frac{\partial w^{j}}{\partial u^{i}} \frac{\partial}{\partial w^{j}}$$.

So the bottom line is, when you change the basis from $$\frac{\partial}{\partial u^{i}}$$ to $$\frac{\partial}{\partial w^{j}}$$, the transformation rule changes the coefficents from $$A^{i} \rightarrow A^{i}\frac{\partial w^{j}}{\partial u^{i}}$$ and $$A$$ is unchanged.