Motivation: To gain a more thorough understanding of how bases transform under a mapping. To increase understanding of general linear algebra concepts.
Consider a lattice, $Λ = \Bbb Zv_1 +\Bbb Zv_2,$ in $\Bbb R^2$ with the standard basis.
Consider as well, a lattice $\zeta,$ in $\Bbb R^2_*$ fixing $(1,1)$ as the origin (to get from $\Lambda$ to $\zeta$ I exponentiated each coordinate). So $(n,k) \in \Lambda$ maps to $(e^n,e^k)\in \zeta.$
What is a basis for $\zeta?$
Is it just $\zeta=\Bbb Zw_1+\Bbb Zw_2$, such that $w_1=(e,0)$ and $w_2=(0,e)?$ Here $e\approx 2.718.$
How about $\zeta = w_1^{\mathbb Z} \odot w_2^{\mathbb Z},$ where $w_1=(a,1)$ and $w_2=(1,b)$ for some $a,b>0; a,b\neq 1,$ and $(x_1,y_1)\odot(x_2,y_2)=(x_1 x_2, y_1 y_2)$ and $w^{\mathbb Z} = \{ w^n \mid n \in \mathbb{Z} \}$?