Understanding Geometric Algebra

Coordinate transformations  39

Traditional World 3.2 (Contravariant and covariant vectors) In mathematics, a

set of e lements that can be added or multiplied by scalars is generally called a vector space

or a linear space, and their elements are called vectors. The set of ma trices or the set of

functions (over some domain) are vector spac e s in this sense, since matrices and functions

can be added or multiplied by scalars. The set of a rrays of numbers such as a

i

and b

i

also makes a vector spac e in this sense. Hence, in tensor calculus, such arrays o f numbers

(whether their alignment is horizontal or vertical does not matter) are called “vectors,” and

we say “vector a

i

” and “vector b

i

.” It appears at first sight that the omission of the basis