[jacobolus]

You have it backwards. To understand the inner product you first need to understand the geometric product. ;-)

More seriously though, “linear algebra” is often used to mean “matrix algebra”, which you do not need to understand the basic concepts of geometric algebra. Coordinate-free concepts in linear algebra are geometric algebra concepts, and can be easily taught in a first course on the subject.

What you do need to do is first learn about Euclidean vectors as displacements of Euclidean points (and have some basic grounding in Euclidean geometry of points and lines and circles), after which you can learn about the geometric product of vectors, and the various kinds of multivectors and derived products (e.g. the inner product) which are produced out of that product.

Students can wait until after they have studied the basic concepts to learn more generically about quadratic forms, arbitrary linear transformations (which can be extended to multivector transformations via the “outermorphism”), and so on. And might never need to get into mathematicians’ more abstract/formal concepts of rings and modules and Lie groups and so on, though if they do want to they’ll have some better examples and better intuition about it.