The following set of points is a lattice in ${\mathbb{R}}^{n}$:

${D}_{n}=$ $({v}_{1},...,{v}_{n})|{v}_{1},...,{v}_{n}\in \mathbb{Z}$ and ${v}_{1}+...+{v}_{n}$ is even}

Find vectors ${x}_{1},...,{x}_{k}$ such that ${D}_{n}=Int({x}_{1},...,{x}_{k})$

My solution:

Firstly, I know that if ${x}_{1},...,{x}_{k}\in {\mathbb{R}}^{n}$, then we define $Int({x}_{1},...,{x}_{k})={t}_{1}{x}_{1}+...+{t}_{k}{x}_{k}$ where ${t}_{1},...,{t}_{k}\in \mathbb{Z}$

So, I have deduced that the vectors ${x}_{1},...,{x}_{k}\in {\mathbb{R}}^{n}$ so that ${D}_{n}=Int({x}_{1},...,{x}_{k})$ are any vectors such that ${t}_{1}{x}_{1}+...+{t}_{k}{x}_{k}=({v}_{1},...,{v}_{n})$ where ${v}_{1}+...+{v}_{n}$ are even.

However, I feel my solution is certainly lacking and I was wondering if anyone could help guide me in the right direction.