Matrices and vector spaces
In so far as vector algebra is concerned (see the summary in Section A.9 of Appendix A), a vector can be considered as a geometrical object which has both a magnitude and a direction, and may be thought of as an arrow fixed in our familiar three-dimensional space. This space, in turn, may be defined by reference to, say, the fixed stars. This geometrical definition of a vector is both useful and important since it is independent of any coordinate system with which we choose to label points in space.
In most specific applications, however, it is necessary at some stage to choose a coordinate system and to break down a vector into its component vectors in the directions of increasing coordinate values. Thus for a particular ...