Is linear algebra more “fully understood” than other maths disciplines?
It's closer to true that all the questions in finite-dimensional linear algebra that can be asked in an introductory course can be answered in an introductory course. This is wildly far from true in most other areas. In number theory, algebraic topology, geometric topology, set theory, and theoretical computer science, for instance, here are some questions you could ask within a week of beginning the subject: how many primes are there separated by two? How many homotopically distinct maps are there between two given spaces? How can we tell apart two four dimensional manifolds? Are there sets in between a given set and its powerset in cardinality? Are various natural complexity classes actually distinct?
None of these questions are very close to completely answered, only one is really anywhere close, in fact one is known to be unanswerable in general, and partial answers to each of these have earned massive accolades from the mathematical community. No such phenomena can be observed in finite dimensional linear algebra, where we can classify in a matter of a few lectures all possible objects, give explicit algorithms to determine when two examples are isomorphic, determine precisely the structure of the spaces of maps between two vector spaces, and understand in great detail how to represent morphisms in various ways amenable to computation. Thus linear algebra becomes both the archetypal example of a completely successful mathematical field, and a powerful tool when other mathematical fields can be reduced to it.
This is an extended explanation of the claim that linear algebra is "thoroughly understood." That doesn't mean "totally understood," as you claim!
The hard parts of linear algebra have been given new and different names, such as representation theory, invariant theory, quantum mechanics, functional analysis, Markov chains, C*-algebras, numerical methods, commutative algebra, and K-theory. Those are full of mysteries and open problems.
What is left over as the "linear algebra" taught to students is a much smaller subject that was mostly finished a hundred years ago.
Journals such as "Linear Algebra and its Applications" still publish papers, so certainly not everything about Linear Algebra is known. It's definitely not exempt from Gödel. However, it is relatively well-understood compared to most other subject areas, and a "typical" (in some sense) not-overly-complicated question has, empirically, a fairly high probability of being answerable without too much difficulty.