Exercise 1, § 2

Let $\varphi$ be the canonical homomorphism of $\mathbf{R}^n$ onto $\mathbf{T}^n$ and let $u$ be a linear mapping of $\mathbf{R}^m$ into $\mathbf{R}^n$. For $\varphi \circ u$ to be a strict morphism of $\mathbf{R}^m$ into $\mathbf{T}^n$ it is necessary and sufficient that either:

a) $\overline{u}^1(\mathbf{Z}^n)$ is of rank $m$; or

b) $u(\mathbf{R}^m) \cap \mathbf{Z}^n$ is of rank equal to the dimension of $u(\mathbf{R}^m)$.