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**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 ...

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