Revisiting some older exercises from a linear algebra book I was reading, I found the following exercise:

b) Find a subspace $S$ of $V$ for which $S$ is also a vector space under different operations

I remember struggling hard to find such subspace (not counting cases where the operation was pretty much the same, like defining transpose of the addition in the subspace of symmetric matrices), besides the obvious case of conjugate scalar multiplication, to the point I gave up. Even now, I'm still struggling to find any such case besides the one on $\mathbb{C}$. Is there any other example on which $S$ is also a vector space under different operations, changing either (or both) scalar multiplication or addition? Many of my attempts were sabotaged by the presence of $0$ in $S$, where $0$ is the additive identity (under the original operation).

I've considered defining multiplication as $\phi(r) v$, but this implies (if I'm not mistaken) that $\phi$ must be a ring endomorphism, and since we must have $\phi(1) = 1$ for this to work, in $\mathbb{R}, \mathbb{Q}$, $\phi$ can only be the identity, and in $\mathbb{C}$ also the conjugate.


Take $V=\Bbb R^2$, with its standard vector space structure over $\Bbb R$. Now, consider the bijection$$\begin{array}{rccc}f\colon&\Bbb R^2&\longrightarrow&\Bbb R^2\\&(x,y)&\mapsto&(x^3,y^3).\end{array}$$Consider on $\Bbb R^2$ the vector space structure such that the sum is $v\oplus w=f\left(f^{-1}(v)+f^{-1}(w)\right)$ and the product by a scalar is $\lambda\odot v=f\left(\lambda f^{-1}(v)\right)$. Then $\Bbb R\times\{0\}$ is a subspace of $\Bbb R^2$ with respect to both structures.


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