Differential Equations and Linear Algebra, 4th Edition

4 Vector Spaces

4.1 The Vector Space R³

Here we take a fresh look—from the viewpoint of linear algebra—at the familiar 3-dimensional space of the physical world around us (and of multivariable calculus). This review, combining old and new ideas, will provide an introduction to basic concepts that are explored further in subsequent sections of the chapter.

We define three-dimensional coordinate space ${\mathbf{\text{R}}}^3$ to be the set of all ordered triples (a, b, c) of real numbers. In ordinary language, the elements of ${\mathbf{\text{R}}}^3$ are called points. The numbers a, b, and c are called the coordinates of the point P(a, b, c) and can be regarded as specifying the location of P in a fixed xyz-coordinate system.

The location of the point P can also be specified by means ...