I can share what I did having a similar concern in mind, but it was for point-set topology, not linear algebra. I am not sure how much of this can be translated to linear algebra, since student's minds are already full of preconceptions about what a vector space, but not about what a topological space is.

After many years of tutoring point-set topology, I observed that students systematically thought of all topological spaces as $\mathbb{R}^n$, and that they always wanted to use balls, even if the topology was non-metrizable. Hence, when I got to teach my own point-set topology course, I tried something a bit radical: I did not talk about metric spaces at all until later in the course.

I started with motivation. On the second day, I defined the notions of topology, homeomorphism (but not continuous function), and convergence of a sequence. Then I did only small finite examples first. I gave the students the following exercise: 1) How many topologies can you define in {0,1,2}; 2) How many of them produce homeomorphic topological spaces?; and 3) In how many of them does the sequence $0,1,0,1,0,1, \ldots$ converge to $2$? Then I made sure to give students enough time (and guidance) to solve this exercise before moving to anything else.

I wanted to force the students to accept the abstract notion of topology and to not be scared by it (and to realize that everything we do in point-set topology is logical). Also, in this example, there is no way a student is going to attempt to use balls (particularly when I have not talked about balls). I think it worked well.