Why we have introduced linear algebra?

Textbooks about linear algebra are trying to introduce a subject which is the result of many years of deep understanding of many branches of mathematics, in the sense that linear algebra is the natural common bed for mathematical constructions that were before unrelated. A book could never cover such a huge area in an introductory chapter. A wild and bold analogy in another field could be asking: why did people invented grammar, while we already had an informal understanding of eachother?

I guess some could trace linear algebra back to the invention of infinitesimal calculus by Leibniz and Newton. When you derive a function at some point, you get the slope of the tangent of the curve at this point: this tangent line is an approximation of the function around the point. This approximation is useful because it is easy to compute. But what if I want to do the same thing for a multivariable function? Then I have partial derivatives, but they are only approximation "in one direction" taken separately. I you want an approximation of your multivariable function "in all directions at one", you need to find a way to put all of these partial approximation together. That is what formalize linear algebra, and more precisely linear maps between linear vector spaces.


In addition to other answers and comments:

  • linear algebra affords having a geometric intuition with things that are not naturally geometric. For instance, with linear algebra, you can deal with spaces of polynomials, of functions... and you can apply the same reasonings with polynomials, functions... in these spaces more or less as if you were dealing with vectors in $\Bbb R^n$. That explains the first difficulty in linear algebra: the definition of a vector space is quite abstract at first sight. But it's precisely that abstraction which explains the great power of linear algebra (and it's often the case in maths in general) because it's that abstraction which affords applying the same language and the same theorems to so different mathematical objects.

  • consequently, linear algebra is part of the common language of all mathematicians. It's quite frequent to hear, at a higher level, that something "is simply linear algebra" to say that what remains to do should be easy for every mathematician listening to this conversation.

Moreover, and simpler than the other points I mentioned: we need more than 2 or 3 dimensions! Many examples:

  • in physics when they add time as a 4th dimension and some advanced theories like string theory use at least 10 dimensions (!) to describe reality

  • in functional analysis, the goal is to study spaces of functions (e.g. spaces of continuous functions on an interval, or differentiable, or integrable, or being the solutions of a particular differential equation or PDE) which can even have infinitely many dimensions!

  • in statistics, you may need more to study than 2 or 3 variables because in real life, a same phenomenon may depend on lot more than 2 or 3 parameters (e.g. weather can depend on temperature, wind speed, pressure, precipitations, humidity...) or you may want to compare more than 2 or 3 parameters (for instance, compare unemployment rate in 20 different countries).


This part from the Wikipedia article on linear algebra is nice:

  • From the study of determinants and matrices to modern linear algebra

It is mostly about starting with the study of solutions of systems of linear equations and continues with the development of matrix notation up to the axiomatic definition of vector spaces.

Another line is analytic geometry, or vector geometry, combining algebra and geometry. See for example:

  • Analytische Geometrie - Geschichte

Nicolas Bourbaki went a step further: he omitted geometric terms like point, line, etc. and thought with the treatment of linear algebra everything necessary was said.

Combined with calculus one arrives at vector calculus, for example, see:

  • Vector calculus

One important extension going toward infinite many dimensions are:

  • Hilbert spaces

An extension with linear inequalities leads to the economic important

  • Linear programming

Tags:

Linear Algebra

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