Definition. A lattice in a finite-dimensional (real) vector space is the set of all linear combinations with integer coefficients of the members of some basis of that vector space.
Note. Even if the vector space is a Euclidean vector space, i.e., it is meaningful to speak of orthonormal bases, one does not necessarily use the word lattice to mean only the set of all linear combinations with integer coefficients of the members of some orthonormal basis. Such a lattice is called rectangular.
Example. The set of all points of R^{n} with integer coordinates is a rectangular lattice. It is sometimes called the standard lattice in R^{n}.
Proposition. If L is a lattice in R^{n}, the set of all translations T_{a} by elements a of L is a transformation group. (Inasmuch as there is a one-to-one correspondence between the members of the lattice and the translations given by them, one sometimes regards a lattice as a transformation group.)
Theorem. The set of all affine transformations of R^{n} for which the standard lattice is an invariant set is the set of all maps x -> M x + c, where M is an n \times n integer matrix with determinant {+/-} 1 and c is a point with integer coordinates. Note: The case where M is an integer matrix with non-zero determinant (rather than determinant {+/-} 1), and where c is an integer point, corresponds to an isometry that carries the standard lattice to a sublattice of itself.
Show that the commutator of two rotations about different centers is a translation.
Let A, B, and C be the vertices of a given triangle in the plane, and let the corresponding lower case letters denote the lengths of the sides of this triangle opposite these vertices. Let P be the point on the side opposite the vertex A where the bisector of the angle at A meets this side. Show that the barycentric coordinates of P with respect to B and C are
One sometimes says that P is the barycentric combination of B and C that is weighted by the lengths of the opposite sides. This is a reference to the fact that the vector (above) of barycentric coordinates is the unique vector parallel to the vector (b, c) for which the sum of its coordinates is 1. Hint: Use the law of sines.
Use the formula of the preceding exercise to determine a similar formula for the barycentric coordinates, with respect to A, B, and C, of the point where the three angle bisectors of the given triangle meet.
Give a construction for the point where the three altitudes of a given triangle meet.
Let A, B, and C be the vertices of a given triangle in the plane, and let the corresponding lower case letters denote the sides of this triangle opposite those vertices.
Show that the axes of the glide reflections sigma_{c} \circ sigma_{b} \circ sigma_{a} and sigma_{c} \circ sigma_{a} \circ sigma_{b} both pass through the foot of the altitude drawn from the vertex C to the side c and have supplementary angles of elevation with respect to c.
Show that the six glide reflections obtained by composing in various ways the three reflections in the sides of the triangle share three axes. Hint: Look at inverses.
Show that the vertices of the triangle formed by the three axes of the six glide reflections in the previous part are the feet of the three altitudes of the triangle.