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?
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)