Vectors and Volumes

Abstract

ℝ³, the set of ordered triples (x₁,x₂,x₃) of real numbers, is a natural and useful model for physical space. Similarly, ℝ⁴ is an obvious model for space-time. More generally, problems in the sciences or engineering that involve N variables are often investigated in the setting of ℝ^N. Such problems often require the Standard ideas of analysis: continuous change, instantaneous rates of change, integration, and so forth. To adapt these concepts from a one dimensional to an N-dimensional setting, it is first helpful to introduce some algebraic structure on ℝ^N, the structure of a vector space, and then to consider transformations of Euclidean N-dimensional Spaces, particularly the simple and very useful ones known as linear transformations.