> The fundamental truth is that matrices represent linear transformations, and all of linear algebra is developed in terms of linear transformations...
Not at all. It is most definitely not a fundamental truth. I think you are conflating matrix multiplication with matrices, as if that's the only operation that one does on matrices.
Matrices are a way to organize data and what operations you define over the matrices define what it represents.
There are other useful operations on matrices, for example Hadamard products, Schur products, Kronecker product, face split products of matrices that finds use in Physics and Engineering (antenna design for one) where the operation does not represent a linear transformation. They can form an algebra and/or algebraic structures different from the very familiar linear algebraic one.
Not at all. It is most definitely not a fundamental truth. I think you are conflating matrix multiplication with matrices, as if that's the only operation that one does on matrices.
Matrices are a way to organize data and what operations you define over the matrices define what it represents.
There are other useful operations on matrices, for example Hadamard products, Schur products, Kronecker product, face split products of matrices that finds use in Physics and Engineering (antenna design for one) where the operation does not represent a linear transformation. They can form an algebra and/or algebraic structures different from the very familiar linear algebraic one.