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?