What does Emil Artin mean when he says:

It is my experience that proofs involving matrices can be shortened by 50% if one throws the matrices out.

I mean I do understand that matrices are really just Linear Transformations in a vector space and this also makes for cool visualizations associated with all of Linear Algebra. But for the sake of performing manipulations and thinking analytically about Linear Algebra, isn't it essential to have Matrices.

If we throw them out, what else can take its place?

  • $\begingroup$ Well, it's not saying to throw them out completely, but that half of the time they aren't really useful in the context that they're presented. The book I was taught linear algebra with, Linear Algebra Done Right, hardly has matrices in it at all. I think very few to no proofs in that entire book make use of matrices, and determinants are offloaded to the last chapter. At the same time, it still has to use matrices- you can't really discuss upper triangular matrices without matrices, but it doesn't mean that proofs regarding them must employ matrices. $\endgroup$ – Ispil Feb 18 '18 at 7:03
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    $\begingroup$ Not claiming to second-guess Artin, but... matrices are just one representation of elements and transformations in a vector space. Many times, going back and using the fundamental defining properties of those elements and transformations, rather than the matrix algebra of their representation, makes proofs both shorter and arguably more meaningful. $\endgroup$ – dxiv Feb 18 '18 at 7:07

A matrix, in the context of linear maps, is just a representation of a linear map with respect to two choices of a basis (in source and target of the map, each). Most statement about such maps, however, -- in particular when they are geometric in nature -- should be inherently independent of such choices.

For this reason, introducing matrices for talking about such properties, introduces artifacts which often obfuscate the underlying ideas. Most of the relevant statements can be expressed by just referring to maps and vectors

(I'm not claiming that this is what Artin had in mind, though)

  • $\begingroup$ Does that mean still formulating the equations as matrix variables, but not explicitly writing it in "table" form? E.g., you'd still write b = Ax, but not explain the exact content of those variables? Or are we talking about not using matrices at all? $\endgroup$ – André Christoffer Andersen Apr 28 at 7:05
  • $\begingroup$ @AndréChristofferAndersen Here, we are really talking about not using matrices at all. As a matter of fact we may still write b = Ax, but the meaning of this is now that a vector bis the image of a vector x under a linear map which is denoted A. I'm not sure what you mean by 'not explain the exact content' -- all entities are well defined here, you just don't choose a specific representation through a basis. Of course the equation is equivalent to a corresponding equation with matrices and vectors represented by $n\times 1$ matrices. $\endgroup$ – Thomas Apr 28 at 16:30
  • $\begingroup$ My first thought was that Emil Artin was suggesting to stop explicitly stating the "exact content" of a matrix, say [[a_11, a_12, ..., a_1n], [a_21, a_22, ..., a_2n], ..., [a_m1, a_m2, ..., a_mn]], and to start to just use the variable representation of a matrix, say A. That would definitely reduce notation with 50%. I understand that's not what your take-away was. What I don't understand is how you reduce your proof by 50% if the notation stays the same. Are there fewer steps in a proof when using a generalisation like a "linear map" instead of a matrix variable like A? $\endgroup$ – André Christoffer Andersen May 7 at 21:39
  • $\begingroup$ @AndréChristofferAndersen Typically, yes. There are fewer steps. If you want to prove something about a linear map using matrices, you have to choose a basis first, then find a representation of the map in that basis, proof what you intended to proof with the objects found that way (which is typically quite messy in matrix notation), verify that the proof in matrix notation acutally proves what you intended to prove and, finally, verify that the proof and result are actually independent of all the choices you made. $\endgroup$ – Thomas May 11 at 6:15

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