Chapter 7 on Vector Spaces

Linear algebra, the study of vector spaces and linear transformations or matrices, began as the study of simultaneous linear equations. It is indispensable for its applications to physics and engineering and a necessary foundation for mathematical studies beyond the calculus. For many, a class in linear algebra is a first encounter with a more abstract approach to mathematics.

The implicit concept of a vector as a directional quantity dates back at least to Archimedes in his treatment of buoyancy and his celebrated law of levers. The study of simultaneous equations, still the core of linear algebra, is even older, dating back to the Babylonians, 4000 years ago. Yet these ideas were not fully integrated until the 19th century and the first formal definition of a vector space was finally provided by Peano in 1888.

Linear algebra, the subject of Chapter 7, provides an ideal case study for a number of reasons. First, it is advanced, college level, mathematics, but not too advanced. It is the first mathematical topic that a mathematics or science major generally encounters after calculus, less specialized, for example, than differential equations. Linear algebra is abstract, but not too abstract. It can be pursued on a very concrete level, but, for that very reason, it can be used to illustrate the value and power of mathematical abstraction.

Secondly, linear algebra illustrates the interplay between the geometric and measurement perspectives in mathematics. This is particularly evident in the relationship between linear transformations (a geometric perspective) and their coordinate expressions as matrices (a system of measurements). Abstract arguments in linear algebra generally involve the geometric perspective for which the choice of a coordinate system (or a set of “basis” vectors) is treated as an omitted measurement.

The measurement-omission principle also provides an illuminating perspective on so-called “quotient vector spaces.” The idea of an algebraic quotient is wider than the concept of a quotient vector space and has fundamental importance in higher mathematics. But the same underlying principle, that of measurement omission, is always involved and it appears most transparently in the study of vector spaces.

Finally, vector spaces illustrate the transition from the concrete to a more abstract perspective. On a concrete level, the space of potential spatial displacements in all directions is obviously different from the set of quadratic polynomials with real coefficients. Yet they are both three dimensional vector spaces that can be put into an exact correspondence; any argument concerning three dimensional vector spaces applies equally to both. To thus focus on mathematical structure is to treat the specific referents of each vector space as structurally irrelevant, as omitted measurements.

Vector spaces and matrices are a conceptual underpinning of our civilization. Chapter 7 highlights the value and nature of mathematical abstraction, the interrelationships between the geometric and measurement perspectives, and the illumination provided by Ayn Rand’s measurement-omission principle.