That's wrong, as Geometric Algebra IS linear algebra. So you first have to understand normal linear algebra (vector spaces, inner product, etc.) and then you can properly understand geometric algebra.
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.
Hmm, that is interesting. Let me cite a book Geometric Algebra for Computer Science:
To understand the structure of the book, you need a better feeling for what geometric algebra is, and how it relates to more classical techniques such as linear algebra.
I followed first chapters of this book and it does not require any knowledge of Linear Algebra.
I read that book as well, it is great for motivation, but mathematically a little sloppy (otherwise you wouldn't come away from it thinking you don't need linear algebra...). It doesn't say so, but it first introduces the linear algebra you need to know: vector spaces, inner products, etc., but without rigorous proofs for basics.
I'm not sure if I would say LA must be learned first, certainly a study of vector spaces and inner product spaces, but not necessarily a complete course in LA. But you will understand each better by understanding the other. If you understand neither, this book covers them together. http://faculty.luther.edu/~macdonal/laga/
I think there is a lot of misunderstanding out there what LA is. For example, LA certainly does not have to include the cross product. Just do introduce the basics of LA like vector spaces, linear mappings, bases, eigenvalues, singular value decomposition, etc. can easily take a semester to grasp properly (if you haven't had any exposure to that sort of thing before).
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.