The points (1, 3), (2, 4),
(–1, 7) all belong to a lattice.
True ; False
True. All the points belong to the integer lattice.
2.
The points (1, 0), (1, 1)
generate (determine) the integer lattice.
True ; False
True. Notice that (0, 1) = (1, 1) – (1, 0). This means that any integer combination of (1, 0) and (1, 1) can be expressed as an integer combination of (1,0) and (0,1).
3.
The points (1, 0), (2, 2) generate the integer lattice.
True ; False
False. No integer combination of the given vectors can give (1, 1) for example.
4.
Give two simple vectors which generate the (equilateral) triangular lattice.
Easiest is (2, 0) and (1, 3). Notice that we are not too interested in the scale of the lattice here.
5.
The vectors (1, 0, 0),
(0, 1, 0) and (1, 1, 0) generate a
3 -dimensional lattice.
True ; False
False. The three given vectors lie in a plane
(z = 0), and so does the lattice they determine.
6.
The equations x' = x + y, y' = x + 2ydetermine an integral unimodular transformation.
True ; False
True. Here ad – bc = 1.2 – 1.1 = 1, and the coefficients are all integers.
3). Notice that we are not too interested in the scale of the lattice here.