“main”2007/2/16page 234iiiiiiiiCHAPTER4Vector SpacesTo criticize mathematics for its abstraction is to miss the point entirely. Abstractionis what makes mathematics work. — Ian StewartThe main aim of this text is to study linear mathematics. In Chapter 2 we studiedsystems of linear equations, and the theory underlying the solution of a system of linearequations can be considered as a special case of a general mathematical framework forlinear problems. To illustrate this framework, we discuss an example.Consider the homogeneous linear system Ax = 0, whereA =1 −122 −243 −36.It is straightforward to show that this system has solution setS ={(r − 2s, r, s) : r, s ∈ R}.Geometrically we can interpret each solution as defining the coordinates of a point inspace or, equivalently, as the geometric vector with componentsv = (r − 2s, r, s).Using the standard operations of vector addition and multiplication of a vector by a realnumber, it follows that v can be written in the formv = r(1, 1, 0) + s(−2, 0, 1).We see that every solution to the given linear problem can be expressed as a linearcombination of the two basic solutions (see Figure 4.0.1):v1= (1, 1, 0) and v2= (−2, 0, 1).234“main”2007/2/16page 235iiiiiiii4.1 Vectors in Rn235x3v2  (2, 0, 1)x2x1v1  (1, 1, 0)v  rv1 + sv2Figure 4.0.1: Two basic solutions to Ax = 0 and an example of an arbitrary solution to thesystem.We will observe a similar phenomenon in Chapter 6, when we establish that everysolution to the homogeneous second-order linear differential equationy+ a1y+ a2y = 0can be written in the formy(x) = c1y1(x) + c2y2(x),where y1(x) and y2(x) are two nonproportional solutions to the differential equation onthe interval of interest.In each of these problems, we have a set of “vectors” V (in the first problem thevectors are ordered triples of numbers, whereas in the second, they are functions thatare at least twice differentiable on an interval I ) and a linear vector equation. Further, inboth cases, all solutions to the given equation can be expressed as a linear combinationof two particular solutions.In the next two chapters we develop this way of formulating linear problems in termsof an abstract set of vectors, V , and a linear vector equation with solutions in V . We willfind that many problems fit into this framework and that the solutions to these problemscan be expressed as linear combinations of a certain number (not necessarily two) of basicsolutions. The importance of this result cannot be overemphasized. It reduces the searchfor all solutions to a given problem to that of finding a finite number of solutions. Asspecific applications, we will derive the theory underlying linear differential equationsand linear systems of differential equations as special cases of the general framework.Before proceeding further, we give a word of encouragement to the more application-oriented reader. It will probably seem at times that the ideas we are introducing are ratheresoteric and that the formalism is pure mathematical abstraction. However, in additionto its inherent mathematical beauty, the formalism incorporates ideas that pervade manyareas of applied mathematics, particularly engineering mathematics and mathematicalphysics, where the problems under investigation are very often linear in nature. Indeed,the linear algebra introduced in the next two chapters should be considered an extremelyimportant addition to one’s mathematical repertoire, certainly on a par with the ideas ofelementary calculus.4.1 Vectors in RnIn this section, we use some familiar ideas about geometric vectors to motivate the moregeneral and abstract idea of a vector space, which will be introduced in the next sec-tion. We begin by recalling that a geometric vector can be considered mathematicallyas a directed line segment (or arrow) that has both a magnitude (length) and a directionattached to it. In calculus courses, we define vector addition according to the parallel-ogram law (see Figure 4.1.1); namely, the sum of the vectors x and y is the diagonal of“main”2007/2/16page 236iiiiiiii236 CHAPTER 4 Vector Spacesthe parallelogram formed by x and y. We denote the sum by x +y. It can then be showngeometrically that for all vectors x, y, z,xyx  yFigure 4.1.1: Parallelogram lawof vector addition.x + y = y + x (4.1.1)andx + (y + z) = (x + y) + z. (4.1.2)These are the statements that the vector addition operation is commutative and associa-tive. The zero vector, denoted 0 , i s defined as the vector satisfyingx + 0 = x, (4.1.3)for all vectors x. We consider the zero vector as having zero magnitude and arbitrarydirection. Geometrically, we picture the zero vector as corresponding to a point in space.Let −x denote the vector that has the same magnitude as x, but the opposite direction.Then according to the parallelogram law of addition,x + (−x) = 0. (4.1.4)The vector −x is called the additive inverse of x. Properties (4.1.1)–(4.1.4) are thefundamental properties of vector addition.The basic algebra of vectors is completed when we also define the operation ofmultiplication of a vector by a real number. Geometrically, if x is a vector and k isa real number, then kx is defined to be the vector whose magnitude is |k| times themagnitude of x and whose direction is the same as x if k>0, and opposite to x ifk<0. (See Figure 4.1.2.) If k = 0, then kx = 0. This scalar multiplication operationhas several important properties that we now list. Once more, each of these can beestablished geometrically using only the foregoing definitions of vector addition andscalar multiplication.xkx, k  0kx, k  0Figure 4.1.2: Scalarmultiplication of x by k.For all vectors x and y, and all real numbers r, s and t,1x = x, (4.1.5)(st)x = s(tx), (4.1.6)r(x + y) = rx + ry, (4.1.7)(s +t)x = sx + tx. (4.1.8)It is important to realize that, in the foregoing development, we have not defined a“multiplication of vectors.” In Chapter 3 we discussed the idea of a dot product and crossproduct of two vectors in space (see Equations (3.1.4) and (3.1.5)), but for the purposesof discussing abstract vector spaces we will essentially ignore the dot product and crossproduct. We will revisit the dot product in Section 4.11, when we develop inner productspaces.We will see in the next section how the concept of a vector space arises as a directgeneralization of the ideas associated with geometric vectors. Before