It can be beautiful with less.

All that needs to be demonstrated is that for real numbers + associates and commutes. That * associates and commutes. And most satisfyingly, these two operations interact through the distribution property.

Of course, it's more revealing and interesting if one has some exposure to groups and fields.

Do people encounter linear algebra in their course work before that ?

For us it came after coordinate/analytical geometry where we had encountered parallelogram law. So while doing LA we had some vague awareness that there's a connection. This connection solidified later.

We also had an alternative curriculum where matrices were taught in 9th grade as a set of rules without any motivation whatsoever. "This is the rule for adding, this one's for multiplication, see you at the test"