Linear Spaces

Definition: Linear Space

Let's begin by abstracting what we have done in the two cases above. The first is the easiest to think about. In the case of describing an object moving in three space, we have started with three objects for which we know some physical properties or have some intuitions about: the three basis directions. We then multiplied each direction be a length and added the resulting three vectors formally. We can abstract what we got as follows. A linear space (or a vector space) is a set of elements (vectors), V, and a set of numbers (scalars), S, (where for us, S will be either the real or complex numbers) satisfying the following properties: (Click here for a definition of the specialized math symbols used.)

These properties specify a linear (or vector) space. It is easy to show that the two examples discussed in the Motivation section both satisfy all these properties.



This page prepared by

Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 405-6120

Last revision 25. October, 2005.