A. Explain the general relationship between abstract algebra and linear algebra.
B. Discuss how it is possible to considerably shorten the list of properties that define a vector space by using definitions from abstract algebra.
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The beginning of linear algebra is the study of solutions of sets of linear equations. The first step is to understand precisely what operations one can do to a set of linear equations that maintain the set of solutions, and that allow us to find the solution by reducing the set of linear equations to a set of solutions. The set of operations that maintain the set of solutions is obviously a group but in reality a fairly complicated one that we understand through operations such as switching the order of equations, adding one equation to another, and multipliying an equation by a nonzero number. It might be interesting to better understand in this fashion what property of this group allows us to solve...
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