In linear algebra, we care a lot about dimensions. I get why it’s useful but not why it’s such a big deal. So I wondered what problem was solved historically by introducing a rigorous definition of dimension of a vector space?

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    $\begingroup$ What makes you think that the definition had to be a response to a problem that needed to be solved? And what makes you think that it was a definition that was created at a discrete moment in time? It was probably a gradual evolution of styles of definition. $\endgroup$ – Ben Crowell May 30 at 14:09

I will begin by 2 examples where the notion of dimension is essential :

  • The increasing sequence of subspaces

$$\{0\}\subset\ker(A-\lambda I)\subset\ker(A-\lambda I)^2\subset\ker(A-\lambda I)^3\subset\dots$$

(with ultimate stabilisation at a certain step) and the connection of their dimensions with the so important Jordan decomposition of a matrix $A$ relatively to eigenvalue $\lambda$ (well described in this answer by Bernard here).

  • the concept of degree of field extensions (see paragraph "examples" here) where for example $\mathbb{Q} (\sqrt {2},\sqrt {3})$ can be considered in two different ways as a vector space of dimension $4$ over $\mathbb{Q}$.

But, on an historical point of view, the concept of dimension finds its roots in the connection with

  • the concept of rank, which was initially defined as the maximal size of a sub-determinant one could extract from a determinant ,[in modern terms : replace in the previous sentence : "determinant" by "matrices"]. We were far, when this concept has emerged, from "rank = dimension of the range of the corresponding linear mapping". This concept of "rank" has emerged (not with this name) when Rouché has published an article in Journal de l'Ecole Polytechnique (1880) explaining the kind of solutions one could expect from a linear system of $n$ equations with $p$ unknowns and rank $r \le \min(n,p)$ .

  • [connected to the previous item] the "rank-nullity" theorem. Mathematicians have been accustomed more and more to identify a subspace either as the kernel or the range of a certain linear mapping. This can be traced back to 1884 where Sylvester has defined the "nullity" of a square matrix

Here is a quotation of https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants/ : Sylvester defined the nullity of $n(A)$ of matrix $A$, to be the largest $i$ such that every minor of $A$ of order $n-i+1$ is zero. He was interested in invariants of matrices, that is properties which are not changed by certain transformations. He proved that $$\max \{n(A), n(B)\} \le n(AB) \le n(A) + n(B)$$.

Remark : The concept of dimension/rank can be enlarged

  • "The rank of a mathematical object is defined whenever that object is free. In general, the rank of a free object is the cardinal number of the free generating subset G" as explained in https://mathworld.wolfram.com/Rank.html.

  • it is also to be connected to the "number of degrees of freedom", a somewhat looser concept that can be much more complicated when you are not in the linear case, with manifolds (generalized surfaces) instead of vector spaces.

  • it has been discovered in the 1960s in the framework of numerical analysis that there is a notion of low rank approximation expressing that for example a set of points in the $n$-dimensional space is "close to belong to $p$-dimensional subspace", using the concept of "singular value decomposition" (a generalization of eigenvalues/vectors decomposition). These concepts are very fundamental in present-day treatments of the so-called "big-data".

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    $\begingroup$ This answer is very thin on historical facts. This seems like a math answer, not a history answer. $\endgroup$ – Ben Crowell May 30 at 14:11
  • $\begingroup$ @Ben Crowell I have attempted to give a broader perspective (in particular more historically grounded) to this answer. $\endgroup$ – Jean Marie Becker Jun 3 at 22:37

The idea of dimension was already intuitively understood. For example, the ancient Greeks distinguished between plane and solid geometry. And early modern physicists used the notion of degrees of freedom.

It was Liebniz who called for an algebra that would treat the figure as directly as it treated quantity which he called analysis situs. In fact, it was on the basis of his theorising that a competition was instituted which was won by Hermann Grassmann. In fact, if I recall correctly it was Mobius that encouraged Grassmann to enter it. This was on the basis of his book, The Theory of Linear Extension: A New Theory of Mathematics, and what now is known as linear algebra, and more precisely as the exterior or Grassmann algebra.

In this theory the intuitive understanding of dimension is given a rigorous foundation. This is simply the cardinality of any basis.

So I was wondered what problem was solved historically by introducing a rigorous definition of dimension.

The historical problem was the one posed by Liebniz: to give a rigorous exposition of an algebra that treats extension directly. This is the theory of vectors and tensors. An outcome of this theory was a rigorous definition of dimension which showed that this theory was on the right track.

I believe Grassmann actually credited his father with the crucial notion that helped him theorise about extension synthetically.

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  • $\begingroup$ I agree that a good answer to the question should probably mention Grassmann (and, by the way, Riemann's habilitation which talked of dimensions of manifolds way before we had the linear concept), but unfortunately what you write there is unclear to me. What "competition" did Grassmann win? $\endgroup$ – Torsten Schoeneberg Jun 3 at 7:08
  • $\begingroup$ @torsten Schoeneberg: I don't recall now. It was referenced in a biography of Grassmann that I read a few months ago. I think it was Mobius that encouraged Grassmann to enter it. As I have already written above, it was a competition to flesh out Liebnizs proposal for an algebra of extension. $\endgroup$ – Mozibur Ullah Jun 3 at 7:53
  • $\begingroup$ I see; yes, wikipedia mentions that prize too. I would think though that Grassmann's main work, the Lineale Ausdehnungslehre, in the two editions published during his lifetime, should foremostly be mentioned here. $\endgroup$ – Torsten Schoeneberg Jun 3 at 21:16

I'm not convinced there was any specific "problem," but simply that a full formal definition allowed people to use multidimensional notation as a shorthand way of defining and managing large amounts of variables. Sort of like working out the rules for exponentiation, for example.

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