Section 15.2 Motivation
The rules of vector algebra listed in Proposition 11.5.1 are valid whether we take a geometric view (using directed line segments) or an an algebraic view (using column vectors) of vector addition and scalar multiplication. But all these rules have counterparts in matrix algebra in Proposition 4.5.1, which suggests that these algebra patterns might be more universal — are there other collections of algebraic objects that can be added and scaled and that follow the same rules of algebra when we do so?
If we observe similar algebraic patterns elsewhere (and we will), then it is worth the effort to abstract the concepts of vector and vector algebra — to disassociate them from any specific ideas of what they are, and deal with them as abstract concepts. This is ultimately where mathematics becomes most powerful: when it recognizes, describes, and analyzes patterns in familiar contexts that can then be recognized and exploited in new contexts.
The cycle of life of a mathematical idea is as follows.
- Extract common patterns from familiar model systems (on the left in Figure 15.2.1 below).
- Describe the core features of these common patterns and use them as the basis for an abstract system.
- Deduce new properties of the abstract system based on the aspects of the underlying patterns that describe it.
- Recognize the common patterns described by the abstract system in new systems (on the right in Figure 15.2.1 below).
- Interpret the new abstract properties back in the known systems, new and old, and apply these properties to solve problems.
Since the abstract properties are logically deduced from the underlying patterns that defines the abstract system, every specific system that follows these common patterns must have the same properties.
Following this cycle for systems that follow the patterns of the rules of algebra for vector addition and scalar multiplication is our task for the next few chapters. In this chapter, we begin our study of the abstract system we can extract from our familiar model systems of vectors in \(\R^n\) and \(m \times n\) matrices, both of which satisfy the same rules of algebra with respect to addition and scalar multiplication.
