Lattice Basis Reduction

Linearly Dependent Vectors 113

that ℓ = 2 does not make any row of the right block equal to zero, but that

ℓ = 3 makes the fir st row of the right block equal to zero.

Exercise 6.4. Use a co mputer algebra system to redo Example 6.1 with the

limiting value α = 1 (β =

4

3

). Show that ℓ = 13 does not make any row of the

right block equal to zero, tha t ℓ = 14 makes the second row equal to zero but

not the first, and that ℓ = 15 makes the first and third rows equal to zero.

Exercise 6.5. Consider these four row vectors in R

3

:









x

1

x

2

x

3

x

4









=









−32 8 44

−74 69 92

−4 99 −31

27 29 67









Following the method of Section 6.1 with α =

3

4

(β = 2), find

(i) the smallest ...