Of course you can do linear algebra without vector spaces. Leibniz didn't know about vector spaces, yet he was doing linear algebra. It just happens that the use of vector spaces massively helps thinking about linear algebra problems.
CT is applied to many domains. For a concrete example look up ZX calculus, which is used to optimize quantum circuits.
> It just happens that the use of vector spaces massively helps thinking about linear algebra problems.
That is the point here.
> CT is applied to many domains.
Yes, but in which of those does it massively help? I just looked up ZX calculus, and I am sure you can formulate that better by not mentioning CT at all.
As usual, quick googling reveals that the ZX calculus is really just a CT repackaging of Penrose diagrams, which were discovered and used without any involvement of CT.
