I need to find an isomorph between the spaces $\Bbb{R}^3$ and vector space with defined operations on $\Bbb{R}^3$:
$$\vec{v} \oplus \vec{w} = \vec v + \vec w - \vec u$$ $$r \odot \vec{v} = r \cdot (\vec v - \vec u) + \vec u $$ Where $\vec u$ is fixed to be $$\vec u = \begin{pmatrix} 2 \\ 0 \\ -1 \end{pmatrix}$$
I'm lost at any way to do this, I tried $$f: \begin{pmatrix} x \\ y \\ z \end{pmatrix} \rightarrow \begin{pmatrix} x-2 \\ y \\ z+1 \end{pmatrix} $$ But scalar multiplication didn't hold, how do I go about finding an isomorph, is there a systematic way? Or do I just keep guessing?