How is linear algebra useful in higher mathematics?

Question

How are vector spaces fundamental to higher mathematics (functional analysis, number theory).I'm familiar with vector spaces, basis vectors, inner products, eigenvalues,groups, rings , fields, modular arithmetic etc. I've read Axler's Linear Algebra Done Right and a bit of group theory.

Answer 1

Here's a few quick examples of applications of linear algebra to higher mathematics. The examples I've picked are often approached in the middle or higher levels of an undergraduate mathematics major, and are certainly part of the general education curriculum of a mathematics graduate student.

The study of linear algebra over the base field $\mathbb Q$ is at the heart of a central branch of number theory known as "algebraic number theory". For example, $\mathbb Q(\sqrt{2}) = \{a + b \sqrt{2} \mid a,b \in \mathbb Q\}$ is a field over $\mathbb Q$ of dimension $2$, and that dimension just so happens to equal the degree of the polynomial $x^2 - 2$. More generally, any field $F$ that contains $\mathbb Q$ as a subfield is also a vector space over $\mathbb Q$, and if $F$ is finite dimensional over $\mathbb Q$ then its dimension is intimately connected with the degrees of polynomials with integer coefficients having roots in $F$.

Linear algebra is the basis of the study of linear differential equations. For example, the solution space of the equation $y''-y=0$ is a 2-dimensional vector space over $\mathbb R$ in which the two functions $e^x$, $e^{-x}$ form a basis (the two functions $\cosh(x)$, $\sinh(x)$ also form a basis!). In general, any homogeneous $n^{\text{th}}$ degree differential equation over $\mathbb R$ has a solution space which is a vector space of dimension $n$, and finding $n$ linearly independent solutions that form a basis for the solution space gives a very concrete description of the solution space.

Linear algebra is useful in the study of topology. In the 19th century, the progenitors of topology like Betti and Poincare used linear algebra to come up with numbers that came to be called "Betti numbers", which in some intuitive sense count the number of "holes" in a topological space. These "Betti numbers" can be used to tell topological spaces apart from each other. For example, the 1st Betti number of the 2-sphere is equal to $0$ and the 1st Betti number of the torus is equal to $2$, which let's us conclude that the 2-sphere and the torus are topologically distinct. As 20th century topology unfolded, the connection to linear algebra was clarified as it was discovered that the $1^{\text{st}}$ Betti number of a topological space was equal to the dimension of an important vector space associated to that space, known as the $1^{\text{st}}$ homology of the space.

Answer 2

Algebra, in school, is not much more than a bunch of rules for solving equations, expanding or factorising expressions and so on. But once you get past school, you find that algebra involves abstract structures like groups, rings, fields and vector spaces.

So, what is algebra in the advanced sense all about? IMHO it is about recognising when apparently different situations are "really the same". Perhaps the very best example of this is group theory, which encompasses a great deal of information about ordinary arithmetic, modular arithmetic, geometric transformations (including relativity), functions, permutations and so on, all of which may very well appear on a superficial level to be entirely separate studies.

Linear algebra is another great example of this kind of unification. Vectors as pairs of numbers, vectors as arrows, polynomials, matrices, functions, sequences, likewise may appear to be completely separate studies, but they can all be regarded as instances of one topic: vector spaces.

OK so we can do all this, but why would we want to? I would suggest two reasons. The first is simply a question of saving ourselves work: it would be crazy to prove a result about (let's say) bases for vectors, then go through all the work again to prove what is essentially the same result for matrices, then again for polynomials and so on. The second is that by recognising connections between apparently different situations, we deepen our understanding of the whole picture.

Answer 3

From a philosophical poit of view, I dare say that Taylor'r expansion is the bread for enginiers and most of laws of phisics are (in first aproximation) "linear" relations betweel cause an effect, more when the low is deduced by experience. So, linear algebra is the bread for math, as a tool that satisfys the pleasure of knowledge. The bread must be placed on a desk before any other thing. Add Stone's theorem on approximation of a continuous function by a polinomy, and you have a complete lunch. The bird is the completion of the field of rationals, which is just a bit more then then the gift of the God.