[Someone]

I think you overestimate "the general layman". For example, complex analysis requires one to accept the existence of the square root of minus one and that multiplying by it is equivalent to a rotation.

In my experience, "the general layman" simply doesn't possess enough suspension of disbelief to get past such hurdles.

[ouid]

linear algebra, though, I feel is an excellent example.

Everyone has solved a system of linear equations in school. The concept is pretty accessible.

[Someone]

In 2D, I'll give you that, but in general, I would say "has struggled solving", rather than "has solved", and not even 'everyone'.

Few can, for example, reliably compute the intersection point of a line l' parallel to l through a point p with a plane P, and many would panic when given a problem in N>3-dimensional space.

I know people who memorized formulas for solving linear systems as used in macro-economics (on the order of 10 dimensions, but lots of zeroes in their matrix), and, because of that, couldn't work when given a slightly different model to solve (for example, if taxation became be a constant plus a fraction of income instead of just a fraction of income, or if capital gains tax were introduced).

[ouid]

What on earth are you talking about?

The first step in a linear algebra course is to codify the method that you use in middle/high school to solve systems of equations of the form a00x0+a01x1+...=b0....

That algorithm sets the stage for everything. The pace is natural, the math is accessible, no geometric interpretation is ever required, and it's built for problems that people of even average intelligence can understand. Frankly, it's astonishing we don't teach it in high school.