Verify if linear combination of vectors is in lattice

Question

Let $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ be vectors in $\Bbb{R}^3$. How do I verify if there is a linear combination of them that belongs in the lattice $\mathcal{L}(B)$ where $B = \{(1,1,1)\}$?

Answer 1

1) Check if your vectors are linearly independent. If they are this becomes trivial and the answer is yes.

2) If the answer is no, reduce $\{a,b,c\}$ to a set of independent vectors $\Gamma$.

pos 1: $\Gamma = \{a\}$. Check if $a$ is of the form $(n,n,n)$.

pos 2: $\Gamma = \{a,b\}$. check if the system of equations $\lambda a + \mu b = (1,1,1)$ has a solution.