It seems like you are confusing vector algebra with linear algebra. The cross product is not a part of the theory of vector spaces and linear transformations.

It seems like you are confusing vector algebra with linear algebra. The cross product is not a part of the theory of vector spaces and linear transformations. Geometric algebra is pretty cool, but it is a potential replacement for vector algebra, not linear algebra. It is defined on a vector space, which means that the theory of linear algebra (concepts like linear transformation, bases, etc.) applies. As a sidenote, reducing Maxwell's equations in the 4-equation Heaviside form to a single equation is not necessarily a boon. Each of the four has a clear physical meaning in the vector calculus formulation, which may be lost when moving to a single equation.