MAS400: Exercises 7

1. Calculate the Gram–Schmidt orthogonalisations of the following. (a) {(1, 1, 1), (1, 2, 3), (1, 4, 9)}.

(b) {(1, 2, 0), (0, 1, 2), (2, 0, 1)}.

2. Calculate using BasisReduction (LLL-)reduced bases for the following lattices.

(a) Λ_{1} = ⟨(3, −5), (7, −11)⟩_{ℤ}.

(b) Λ_{2} = ⟨(12, 2), (13, 4)⟩_{ℤ}.

(c) Λ_{3} = ⟨(1, 1, 1), (1, 1, −1), (1, −1, 1)⟩_{ℤ}.