# Lattice

Geometry Level 4

A lattice $$\Lambda$$ is a collection of vectors that can be constructed by adding integer multiples of two (linearly independent) basis vectors: $\Lambda = \{ n\vec a + m\vec b\ |\ n, m \in \text{IN} \}.$ Consider the lattice generated by the basis vectors $$\vec a = (2, -1)$$ and $$\vec b = (8, 3)$$, i.e. $\Lambda = \left\{ \left(\begin{array}{c} 2n + 8m \\ -n + 3m \end{array}\right)\ |\ n, m \in \text{IN}. \right\}.$ Which of the following set of basis vectors $$\vec a', \vec b'$$ describes the same lattice?

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