Linear Algebra

Abstract

An algebraic approach to problems of plane geometry led us in §IV.7 to introduce two-dimensional vector spaces over the field R of real numbers, while in the calculus gradients and tangent lines lead to cotagent and tangent vector spaces. Three dimensional vector spaces are standard in physics, while the algebra of vectors is an effective way of handling geometrical ideas in dimensions higher than 3. Analysis soon produces infinite-dimensional spaces such as L² (§VI.11). This chapter will summarize the properties of such linear vector spaces over an arbitrary field, not necessarily R or C.