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32 Chapter One. Linear SystemsIILinear Geometry of n-SpaceFor readers who have seen the elements of vectors before, in calculus or physics,this section is an optional review. However, later work will refer to this materialso it is not optional if it is not a review.In the first section, we had to do a bit of work to show that there are onlythree types of solution sets — singleton, empty, and infinite. But in the specialcase of systems with two equations and two unknowns this is easy to see. Draweach two-unknowns equation as a line in the plane and then the two lines couldhave a unique intersection, be parallel, or be the same line.Unique solutionNo solutionsInfinitely manysolutions3x + 2y = 7x − y = −13x + 2y = 73x + 2y = 43x + 2y = 76x + 4y = 14These pictures don’t prove the results from the prior section, which apply toany number of linear equations and any number of unknowns, but nonethelessthey do help us to understand those results. This section develops the ideasthat we need to express our results from the prior section, and from some futuresections, geometrically. In particular, while the two-dimensional case is familiar,to extend to systems with more than two unknowns we shall need some higherdimensionalgeometry.II.1 Vectors in Space“Higher-dimensional geometry” sounds exotic. It is exotic — interesting andeye-opening. But it isn’t distant or unreachable.We begin by defining one-dimensional space to be the set R 1 . To see thatdefinition is reasonable, draw a one-dimensional spaceand make the usual correspondence with R: pick a point to label 0 and anotherto label 1.0 1Now, with a scale and a direction, finding the point corresponding to, say +2.17,is easy — start at 0 and head in the direction of 1 (i.e., the positive direction),but don’t stop there, go 2.17 times as far.
Section II. Linear Geometry of n-Space 33The basic idea here, combining magnitude with direction, is the key to extendingto higher dimensions.An object comprised of a magnitude and a direction is a vector (we will usethe same word as in the previous section because we shall show below how todescribe such an object with a column vector). We can draw a vector as havingsome length, and pointing somewhere.There is a subtlety here — these vectorsare equal, even though they start in different places, because they have equallengths and equal directions. Again: those vectors are not just alike, they areequal.How can things that are in different places be equal? Think of a vector asrepresenting a displacement (‘vector’ is Latin for “carrier” or “traveler”). Thesesquares undergo the same displacement, despite that those displacements startin different places.Sometimes, to emphasize this property vectors have of not being anchored, theyare referred to as free vectors. Thus, these free vectors are equal as each is adisplacement of one over and two up.More generally, vectors in the plane are the same if and only if they have thesame change in first components and the same change in second components: thevector extending from (a 1 , a 2 ) to (b 1 , b 2 ) equals the vector from (c 1 , c 2 ) to (d 1 , d 2 )if and only if b 1 − a 1 = d 1 − c 1 and b 2 − a 2 = d 2 − c 2 .An expression like ‘the vector that, were it to start at (a 1 , a 2 ), would extendto (b 1 , b 2 )’ is awkward. We instead describe such a vector as(b1 − a 1b 2 − a 2)so that, for instance, the ‘one over and two up’ arrows shown above picture thisvector. ( 12)
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