Lattice in $\Bbb R^2$

Question

Let $(a,b)$ be a lattice basis of a lattice $L$ in $\mathbb{R}^2$. Prove that every other lattice basis has the form $(a',b')=(a,b)P$, where $P$ is a $2\times 2$ integer matrix with determinant $1$ or $- 1$.

Answer 1

Since (a,b) and (a',b') are both basis, there exist matrices(2x2) V and W s.t. (a, b) = (a', b') * W, (a', b') = (a, b) * V, and each elements of V and W is integer. (Any basis can create every elements in the lattice)

Combining, (a, b) = (a, b) * VW. And (a, b)*(I-VW) = 0. Yet a and b are linearly independent, thus the matrix (a, b) is non-singular. Now I = VW. 1 = det(I) = det(V)det(W). Because det(V) and det(W) should be integers, det(V) = 1 or -1. From the initial condition, (a', b') = (a, b) * V. It completes the proof.