Differential Equations and Linear Algebra, 4th Edition

4.2 The Vector Space Rⁿ and Subspaces

In Section 4.1 we defined 3-dimensional space ${\mathbf{\text{R}}}^3$ to be the set of all triples (x, y, z) of real numbers. This definition provides a mathematical model of the physical space in which we live, because geometric intuition and experience require that the location of every point be specified uniquely by three coordinates.

In science fiction, the fourth dimension often plays a rather exotic role. But there are common and ordinary situations where it is convenient to use four (or even more) coordinates rather than just two or three. For example, suppose we want to describe the motion of two points P and Q that are moving in the plane ${\mathbf{\text{R}}}^2$ under the action of some given physical law. (See Fig. 4.2.1.) In order to ...