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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