The rules of vector algebra listed in Proposition 12.5.1 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.
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.
A schematic diagram outlining the cycle of abstract modelling in mathematics. Several boxes labelled “specific system” are vertically stacked both on the left and the right of the diagram. The “specific system” boxes on the left feed into a central box labelled “abstract system” via arrows labelled “same patterns,” which in turn feeds into the “specific system” boxes on the right, also via arrows labelled “same patterns.” Finally, the “abstract system” box feeds into a box labelled “properties” which appears below it, via an arrow labelled “logic”. This “properties” box, in turn, feeds back into all of the “specific system” boxes on both sides of the diagram via unlabelled arrows.
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.
