linear algebra   Vectors and vector spaces   Vectors are often
expressed using coordinates. For example, in two dimensions a
vector can be defined by a pair of coordinates (a1a2) describing
an arrow going from the origin (0, 0) to the point (a1a2). If
one vector is (a1a2) and another is (b1b2), then their sum is
(a1 + b1a2 + b2); this gives the same result as the
parallelogram (see the Vectors can be added together by first
placing their tails at the origin of a coordinate system such
that their lengths and directions are unchanged. Then the
coordinates of their heads are added pairwise; e.g., in two
dimensions, their x-coordinates and their y-coordinates are
added separately to obtain the resulting vector sum. As shown by
the dotted lines, this vector sum coincides with one diagonal of
the parallelogram formed with the original vectors.Encyclopædia
Britannica, Inc.). In three dimensions a vector is expressed
using three coordinates (a1a2a3), and this idea extends to any
number of dimensions.   Linear transfo