<p>An affine space is a set consisting of points and vectors. A vector space (or linear space) is an affine space with a specified origin. For a given affine coordinate system, there exists a one-to-one correspondence between vectors and coordinates. In order to provide a theoretical basis for coordinate and frame transformations, and to simplify the transformation process in specific problem studies, the geometric product of vectors is redefined (the geometric product of vectors is the sum of the negative inner product and the outer product of the vectors), thus the one-to-one correspondence relationship between unit vectors and imaginary units is established. According to Hurwitz’s theorem, the vector space of the outer product is gotten and defined, and its dimension cannot be chosen arbitrarily. Based on Arthur Cayley’s (1845) multiplication rules of octonions, transformation formulas for a seven-dimensional vector space and rotation matrices for coordinate frame transformations are derived. It is pointed out that the three-dimensional rotation matrices commonly used in astrometry and geodesy are special cases thereof. According to the redefinition of the geometric product of vectors, the multiplication rules of quaternions can be directly obtained, as well as the multiplication table of imaginary units of octonions. It is further indicated that the multiplication tables of imaginary units for octonions and hypercomplex numbers of higher dimensions are not unique.</p>