I'm reading Fundamental Problems of Algorithmic Algebra by Chee-Keng Yap and could not solve the following problem.
The shortest k-sequence (where k > 2 is the dimension of the lattice) need not form a basis of the lattice.
With following definition of a shortest k-sequence:
We define $u ∈ Λ$ to be a shortest vector in $Λ$ if it has the shortest length among the non-zero vectors of $Λ$. More generally, we call a sequence $(u_1 , u_2 , . . . , u_k )$ for $k ≥ 1 $, of vectors a shortest k-sequence of $Λ$ if for each $i = 1, . . . , k$; $u_i$ is a shortest vector in the set $Λ \setminus Λ(u_1 , u_2 , . . . , u_{i−1} )$.