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- Thread starter TheBiologist
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Stephen Tashi

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No courses, really, apart from school, as I'm still in Secondary School...

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I'm still in Secondary School, so no courses, apart from school itself.

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Stephen Tashi

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Can we assume you have done some simple algebra problems, like solving 3x + 1 = 7 ?

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Yes. I've done indices 2, nth terms, volume, surface area, sequences and we've just started loci.Can we assume you have done some simple algebra problems, like solving 3x + 1 = 7 ?

- #7

HallsofIvy

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"Linear algebra" is a subset of "abstract algebra" but important enough to be considered separately. It looks specifically at those operations we consider "linear". Again, the equation Stephen Tashi gives is linear. The only operations we need to consider are adding and subtracting and multiplying by

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Thank you for the explanation.

"Linear algebra" is a subset of "abstract algebra" but important enough to be considered separately. It looks specifically at those operations we consider "linear". Again, the equation Stephen Tashi gives is linear. The only operations we need to consider are adding and subtracting and multiplying bynumbers. That would include, say 3x+ 4y but not [itex]x^2[/itex], [itex]y^2[/itex], or [itex]xy[/itex].

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Stephen Tashi

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Suppose a mathematician says "Let's forget all the facts of specific problems. Let's focus on the symbols. We will state the facts in symbols and give some rules written in symbols and if you want to proof some statement in symbols is correct, you must do so according to those rules. You can't base your thinking on interpreting the symbols as specific things like spheres or numbers."

That is an exaggeration of what "abstract algebra" is. It describes what would be involved if you were writing computer programs to automatically manipulate symbolic expressions. In actual abstract algebra, you use symbols and apply rules expressed in terms of symbols, but you don't loose sight of some underlying meaing for the symbols. However the symbols usually don't represent things as specific as spheres or forces.

There are introductions to "group theory" that are written for secondary students. If you read those, you can get an idea of the kind of abstract thing I'm talking about. As a simple example, consider the set S of numbers {1,2,3,4}. As mathematics goes, that is a very "concrete" and familiar thing. Now consider the set G of all functions that map the set S onto itself. (Do you understand the terminology "map S onto itself"?) The symbol G denotes something more abstract and, to most people, less familiar. You could get even more abstract by considering the set A of functions that map G onto itself. (And such things are done in abstract algebra.)

The term "linear" has different meanings in various branches of mathematics. I can't state a "global" meaning for it precisely. Speaking imprecisely, "linear" refers to things whose behavior can be expressed in symbols to obey the distributive law: a ( B + C) = aB + aC and the commutative law of addition : B + C = C + B.

An example of such things are the vectors that represent forces. For vectors the "+" symbol represents addition by the use of the parallelogram law. The multiplication of a number times a vector means to stretch the vector's length by a factor determined by the number.

Linear algebra focuses on working with certain "linear" mathematical things: "systems of linear equations", "linear transformations", and vectors. "Vectors" in linear algebra are usually treated in two phases. They are treated as a set of things with abstract symbolic rules. and, in practical applications, they are treated as n-tuples of numbers. (They aren't usually treated as specific physical quantities such as forces , unless the course is taught by the physics department.)

In a college curriculum, people who major in mathematics need to be taught the skills of reading and writing proofs. The first course in abstract algebra or linear algebra is often where this introduction is given.

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