by jinydu » Wed Dec 21, 2005 10:37 pm
Perhaps a quote from my new Linear Algebra textbook ("Linear Algebra with Applications" 3rd edition, by Otto Bretscher) would be helpful:
"Definition: Dimension
Consider a subspace V of R^n. The number of vectors in a basis of V is called the dimension of V, denoted by dim(V).
This algebraic definition of dimension represents a major advance in the development of linear algebra, and indeed of mathematics as a whole: It allows us to conceive of spaces with more than three dimensions. This idea is often poorly understood in popular culture, where some mysticism still surrounds higher-dimensional spaces. The German mathematician Hermann Weyl (1855-1955) puts it this way: "We are by no means obliged to seek illumination from the mystic doctrines of spiritists to obtain a clearer vision of multidimensional geometry" (Raum, Zeit, Materie, 1918).
The first mathematician who thought about dimension from an algebraic point of view may have been Frenchman Jean Le Rond d'Alembert (1717-1783)...."