Example of a group action G on a vector space V that fails to be linear (i.e. fails to be a linear representation). I have seen the following definition of a linear group representation from C. Lent's notes on Representation Theory:

A linear representation $ρ$ of $G$ on a complex vector space $V$ is a set-theoretic
  action on $V$ which preserves the linear structure, that is:
  
  
*
  
*$ρ(g)(v_1 + v_2) = ρ(g)v_1 + ρ(g)v_2, ∀v_1,v_2 ∈ V$
  
*$ρ(g)(kv) = k · ρ(g)v, ∀k ∈ C, v ∈ V$
  

This definition would imply that there exist actions of $G$ on a vector space $V$ which fail the preserve the linear structure of $V$ in this sense.  Can anyone provide a good example? 
 A: After some more thinking I have thought of an example:
Consider the translation action of the integers $\mathbb{Z}$ as a group under additon on $\mathbb{R^2}$ given by:  $n * (x,y) = (x+n, y)$.
We can verify that this does indeed form a group action:


*

*$m * (n * (x,y)) = m * (x + n, y) = (x + n + m, y) = (m+n) * (x,y)$

*$0 * (x,y) = (x + 0, y) = (x,y)$


However, this action does not respect scalar multiplication for $k \neq 0$ and $n \neq 0$ since this gives $n * (k(x,y)) = n * (kx, ky) = (kx + n, ky) \neq (kx + kn, ky) = k(x+n, y) = k (n * (x,y))$.
A: If I remember rightly, a standard sort of example would be: take the space of matrices
$$ A(k)=\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}, $$
for real $k$. This is a group under multiplication with $A(k)A(-m) = A(k-m)$. Define an action of $\rho$ on $\mathbb{R}$ by
$$ \rho(A(k))(v) = v+k $$
for any $v \in \mathbb{R}$. Now, this is an action since
$$\rho(A(k))(\rho(A(m))(v)) = (v+m)+k = v+(k+m) = \rho(A(k)A(m))(v) \quad \text{and} \quad \rho(A(0))(v)=v,$$
but
$$ \rho(A(k))(u+v) = u+v+k \neq u+k+v+k = \rho(A(k))(u)+\rho(A(k))(v), $$
and
$$ \rho(A(k))(\lambda v) = \lambda v+k \neq \lambda(v+k) = \lambda \rho(A(k))(v), $$
so none of the linear structure remains.
A: An important example coming from representation theory is the so called "dot" action of a Weyl group for a semisimple Lie algebra acting on the dual Cartan.  It is given by the formula $w \cdot \lambda = w(\lambda+\rho)-\rho$ where the non-dot action is a linear one, so it can be thought of as a shifted linear action where we have moved the origin to $-\rho$.
A: Here are some examples:


*

*The most extreme example is the symmetric group on the underlying set of $V$.

*The affine group of $V$, which is generated by invertible linear maps and translations. This can be generalized by replacing ${\rm GL}(V)$ with any matrix subgroup $G$ to get $V\rtimes G$. In particular $G=1$ gives us the group $V$ acting on itself by translations.

*Considering $\Bbb R^n$ as a real $n$-dimensional Riemannian manifold, consider


*

*The isometry group of invertible distance-preserving maps.

*The conformal automorphism group of invertible angle-preserving maps.

*The diffeomorphism group of self-diffeomorphisms.

*The homeomorphism group of self-homeomorphisms.



We can check these all act nonlinearly in one go: each action is transitive, hence there are group elements that send the origin to nonzero vectors.
The answers of Chappers, Nate and GoatsRule all fall under the second bullet point.
