48 1. Various Ways of Representing Surfaces and Examples In the case R × R = R2, this corresponds to the fact that the com- position of a flip about a vertical line with a flip about a horizontal line is equivalent to rotation by π around the intersection of the two lines. With the geodesic flip defined, we can then ask whether the prod- uct space X × Y is symmetric, and it turns out that if X and Y are both symmetric spaces, so is their direct product X × Y . In this manner we can obtain many higher-dimensional examples, and so if we were to attempt to classify such spaces, we would want to focus on those which are irreducible in that they cannot be decomposed as the direct product of two lower-dimensional spaces, since the other examples will be built from these. The direct product provides a common means by which we de- compose objects of interest into simpler examples in order to gain a complete understanding. We find many examples of this in linear algebra, in which context the phrase direct sum is also sometimes used. Any finite-dimensional vector space can be written as the di- rect product of n copies of R this is just the statement that any finite-dimensional vector space has a basis. A more sophisticated ap- plication of this process is the decomposition of a linear transforma- tion in terms of its action upon its eigenspaces, so that a symmetric matrix can be written as the direct product of one-dimensional trans- formations, while for a general matrix, we have the Jordan normal form. This process is also used in the classification of finitely generated abelian groups, where we decompose the group of interest into a direct sum of copies of Z and cyclic groups whose order is a power of a prime, so that no further decomposition is possible. Thus the natural counterpart to the study of how a particular sort of mathematical structure can be decomposed is the study of what instances of that structure are, in some appropriate sense, irreducible.
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