Are matrices isomorphic to systems of linear equations?

I know that linear transformations are isomorphic to matrices, in the sense that a matrix can represent a single linear transformation and vice-versa, and they share properties and operations (like composition).

But I think that matrices are also isomorphic to systems of linear equations. See, a system of linear equations can be represented by a matrix. Two equivalent linear systems are represented by two matrices that are line-equivalent. You can convert a linear system into a matrix, than operate on that matrix by applying line-operations, and then convert the resulting matrix back to a linear system which is equivalent (has the same solution) as the original one.

So, are matrices isomorphic to systems of linear equations?

• Part of any reasonable definition of "isomorphic" is that if $A$ is isomorphic to $B$, then $B$ is isomorphic to $A$; this is already contained in your "and vice-versa." That being said, isomorphism is a precise mathematical term with a very specific meaning, so it's not really applied properly here.
– user296602
Commented Oct 23, 2018 at 17:50