Assume that $P\to M$ is a principal $G$-bundle where $G$ is some (compact) Matrix group. Let $\rho\colon G \to \operatorname{Gl}(\mathbb{R}^n)$ be the tautological representation and $\rho^\prime\colon G\to \operatorname{Gl}(V)$ some other representation.


Let
$$ E = P \times\_{\rho}\mathbb{R}^n, \qquad \text{and} \qquad F = P\times\_{\rho^{\prime}}V
$$
be the associated vector bundles.


Using the Chern-Weyl theory and the usual definitions can associate characteristic classes to $P$, $E$ and $F$.


Are they equal?


To make this less vague, assume $G$ is a complex Matrix group and let`s look at $c\_1(P)$, $c\_1(E)$ and $c\_1(F)$. I assume that $c\_1(P) = c\_1(E)$. Is this also true for $c\_1(F)$?


To make it even less vague, consider the Hopf bundle $\mathbb{S}^3 \to \mathbb{S}^2$, and let $\rho\_n\colon U(1)\to \operatorname{Gl}(\mathbb{C})$ be the usual irreducible representations for $n\in\mathbb{Z}$. Then $c\_1$ of these vector bundles are a complete invariant. Assuming $c\_1(\operatorname{Hopf}) = -1$, what is
$$c\_1\left(\operatorname{Hopf}\times\_{\rho\_n}\mathbb{C}\right)?
$$


I apologize if this is too trivial.



Associated vector bundles and Characteristic Classes
The first Chern class of the dual $-L$ of a line bundle $L$ is the negative of the first Chern class of $L$. In general Chern classes of different associated vector bundles are unrelated, and when they are related the story is complicated. In your example, when you tensor line bundles, the first Chern class scales.


