  An introductory overview of vector spaces, algebras, and linear geometries
over an arbitrary commutative field is given. Quotient spaces are emphasized
and used in constructing the exterior and the symmetric algebras of a vector
space. Affine geometries are introduced and generalized by projective
completion. General projective geometries are briefly introduced. Tensor
products and multilinear functions are treated. The exterior algebra of a
vector space and that of its dual are used in treating linear geometry and
Grassmann's regressive product is treated. Scalar product spaces,
orthogonality, and the Hodge star based on a general basis are covered.
