Linear algebra: Modules
Chapters 6, 7, and 10 are devoted to material mathematicians call linear algebra. Here are developed the elementary properties of modules over rings and modules over fields (vector spaces). This includes a study of matrices as they arise in linear algebra. We use at the beginning for examples and motivation spaces of functions. We feel it is a natural intuitive way of approaching the vector space concept for beginners and ties in neatly with coordinate ideas. We have not treated modules over division rings which are not commutative but the experience gained here by the reader should allow him to handle that variation should he meet it. We have interpolated in Section 6.2 and later in the chapter discussions relating vector spaces and modules on the abstract level with vector spaces as directed line segments as used on the more intuitive level in engineering and physics classes. Hopefully this will assist the beginning student in seeing that the subjects are the same.
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