Linear and Abstract Algebra: What Is It?

[TheBiologist]

Not quite sure, could someone kindly explain the concept of the topic to me? Thanks, it really means a lot. :)

 

[Stephen Tashi]

That's asking a lot! (We should get you to explain "the concept" of biology.) Before we try this explanation, could you tell us you background in mathematics? What courses have you already taken?

 

[TheBiologist]

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

 

[TheBiologist]

That's asking a lot! (We should get you to explain "the concept" of biology.) Before we try this explanation, could you tell us you background in mathematics? What courses have you already taken?

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

 

[Stephen Tashi]

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

 

[TheBiologist]

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

Yes. I've done indices 2, nth terms, volume, surface area, sequences and we've just started loci.

 

[HallsofIvy]

"Abstract algebra" is essentially the study of "finite operations"- that is, operations like you do in solving simple equations (like the one Stephen Tashi gives). It includes all arithmetic operations but in a more "abstract" way- looking at general properties rather than specific numbers or operations.

"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 numbers. That would include, say 3x+ 4y but not x^2, y^2, or xy.

 

[TheBiologist]

"Abstract algebra" is essentially the study of "finite operations"- that is, operations like you do in solving simple equations (like the one Stephen Tashi gives). It includes all arithmetic operations but in a more "abstract" way- looking at general properties rather than specific numbers or operations.

"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 numbers. That would include, say 3x+ 4y but not x^2, y^2, or xy.

Thank you for the explanation.

 

[Stephen Tashi]

In solving mathematical problems, people use symbols to represent things. In pure mathematics, people tend to use single letters. In computer programming, whole words are used. If you are thinking about a specific problem then what you write down in symbols is guided by your knowledge of the problem. If you wrote down a symbolic expression and someone claimed it was wrong, you could argue about it using facts about spheres, numbers, forces etc.

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.