Question
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.
Solution
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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