UNIT VECTOR SPACES Structure 3.1 Introduction Objectives 3.2 What Are Vector Spaces? 3.3 Further Properties of a Vector Space 3.4 Subspaces 3.5 Linear Combination 3.6 Algebra of Subspaces Intersection Sum Direct Sum 3.7 Quotient Spaces Cosets The Quotient Space 3.8 Summary 3.9 Solutions/Answers 3.1 INTRODUCTION In this unit we begin the study of vector spaces and their properties. The concepts that we will discuss here are very important, since they form the core of the rest of the course. In Unit 2 we studied R2and R3. We also defined the two operations of vector addition and scalar multiplication on them along with certain properties. This can be done in a more general setting. That is, we may start with any set V (in place of R2or R" and convert V into a vector space by introducing "addition" and "scalar multiplication" in such a way that they have all the basic properties which vector addition and scalar multiplication have in R2and R3.We will prove a number of results about the general vector space V. These results will be true for all vector spaces -no matter what the elements are. To illustrate the wide applicability of our results, we shall also give several examples of specific vector spaces. We shall also study subsets of a vector space which are vector spaces themselves. They are called subspaces. Finally, using subspaces, we will obtain new vector spaces from given oncs. Since this unit forms part of the backbone of the course, be sure that you understand each concept in it. Objectives After studying this unit, you should be able to define and recognise a vector space; give a wide variety of examples of vector spaces; determine whether a given subset of a vector space is a subspace or not; explain what the linear span of a subset of a vector space is; differentiate between the sum and the d~rectsum of subspaces; define and give examples of cosets and quotient.spaces. 3.2 WHAT ARE VECTOR SPACES? You have already come across the algebraic structure called a field in Unit 1. We now build another algebraicstructure from a set, by defining on it the opehtions of addition and multiplication by elements of a field. This is a vector space. We give the definition of a vector space now. As you read through it you can keep in mind the example of the vector space R' over R (Unit 2). d F if it has two operations, namely addition (denoted by +) and multiplication of elements of V by elementsof F (denoted by .), such that the following properties hold : VS1) + is a binary operation, i.e., u + v E V $L u, V-E V. VS~) + is associative. i.e., (U+ v) + w = u + (v + w) + u, v, w E V. VS3) V has an identity element with respect to +, i.e., 3O~VsuchthatOfv = v =v +O$Lv€V. VS4) Every element of V has an inverse with respect to + : For every u E V, 3 v E V such that u +v = 0. v is called the additive inverse of u, and is written as -u. VS5) + is commutative, i.e., u + v = v + u +u, v E V. VS6) . : F x V+V : .(a,v) = a.v is a well defined operation, i.e., $La€Fandv€V,a.v€V. VS7) +a E F and u, v E V, a. (u+v) = a.u + a.v. VS8) $L a,p E F and v E V, (a+P). v = a.v + P.v VS9) +a,p E F and v E V, (a p). v = a.(P.v) VS10) 1.v = v, for all v E V. When V is a vector space over R, we also call it a real vector space. Similarly, if V is defined over C, it is also called a complex vector space. The product of a E F and v E V, in the definition, is often denoted by av instead of a.v. Note that this product is a vector. This operation is called scalar multiplication, because the elements of F are called scalars. Elements of V are called vectors. Now that the additive inverse ot a vector is defined (in VS4), we can give another Definition: If u,v belong to a vector space V, we define their difference u - v to be For example, in R*we have (3.5) - (1,O) = (33) + (-1,O) = (2,5). After going through Unit 2 and the definition of a vector space it must be clear to you that R~ and R~,with vector addition and scalar multiplication, are vector spaces. We now give some more examples of vector spaces. Example 1: Show that R is a vector space over itself. Solution: '+' is associative and commutative in R. The additive identity is O and the additive inverse of x E R is -'x. The scalar multiplication is the ordinary multiplication in R, and satisfies the properties VS7-VS10. Example 2: For any positive integer n, show that the set R" = {(x,, x2, .......,xn) / xi E R} is a vector space over R, if we define vector addition For any field F, and scalar multiplication as : Fn= {(xl ......x,)jx, E F}. Every element of is called an (xi. xZ........... xn) + (Y,,~2, ....... , yn) = (XI + ~1.~1 + yz, ..........., xn + yn), and n-tuple of elements of F. a(xl,x?, ....... x,) = (ax,, ax2, ..... ax,), a E F. Solution: The properties VSl - VSlO are easily checked. Since '+' is associative and commutative in R, you can check that '+' is associative and commutative in Rn also. Further, the identity for addition is (0.0, ........ , O), because (x,,X7, ............, X,) + (0, 0, .............0) - (x, + 0, X2 + 0%........ X,, + 0) = (x,,XI, ...., X,). The additive inverse of (xi, .......... , x,) is (-x , , ...... AX"), Fora, p~R,(a+ p) (x,. ........, x,,) = ((a+ P)x,. ........(a+ p)~,) = (axI + Px,, ........ axn + ax,,) + (Px,. ...... px,) = CY(X,,....... XI,) + P(x[........... XIl) Define scalar multiplication as follows: Vector Spaces For a E R, f E S, let a f be the function given by (af) (x) - :.f(x) %' x E R. Show that S is a real vector space. Solution: The properties VSI - VS5 are satisfied. The additive identity is the function o (XI SUC~that o (x) = o tor all xe R. The inverse off is -f where (-f) (x) = - [f(x)] W x E R. A Example 6: Lt V c R%e given by V = {(x,y) / x.,y E R and y = 5x) We define addition and scalar multiplication on V to be the same as in R2, i.e., (xlt~l)+ (~29~2)=(XI + X27Yl + y2)and a (x,y) = (ax, cry), for a E R. Show that V Fs a real vector space. Solution: First.note that addition is a binary operation on V. This is because (XI,YI) E V, (~2,~2) E V * YI = 5~1.~2= 5x2 3 YI + Y2 = 5(~1+ ~2) - (x,+x,, Y,+Y~)E V. The addition is also associative and commutative, since it is so in R*. Next, the additive I I identity for R2, (0,0), belongs to V and is the additive identity for V. Finally, if 1 (x,y) E V (i.e., y = Sx), then its additive inverse -(x,y) = (-x, -y) E R2. i Also -y = 5(-x). SOthat, -(x,Y) E V. That is, (;i,yj~V --' - (x,y) E V. Thus, VS1 - VS5 are satisfied by addition on V. As for scalar multiplication, if a E R and (x,y) EV,then y = 5x, so that ay = 5(ax). .'., a (x,y) E V. That is, VS6 is satisfied. The properties VS7 - VSlO also hold good, since they do so for R2 Thus V becomes a real vector space. Check your understanding of vector spaces by trying the following exercises. 1 E E3) Let V be the subset of complex numbers given by V = {x + ix( x E R). L Show that, under the usual addition of complex numbers and scalar multiplication defined by a(x + ix) = ax + i (ax), V is a real vector space. I Ver'tor Spaces E7) Show that.Cn is a complex vector space. Note: We often drop the mention of the underlying field of a vector space if it is . understood. For example, we mav say that "Rn is a vector space" when we mean that "Rn is a vector space over R". Now let us look more closely at vector spaces. The examples and exercises in the last section illustrate different vector spaces. Elements of a vector space may be directed line segments, or ordered pairs of real numbers, or polynomials, or functions. The one thing that is common in all these examples is that each is a vector space; in each there is an addition and a scalar E10) Prove that -(-u) = u, 3' u in a vector space. Vector Spaces El 1) Prove that a(u-v) = cxu -av tor all scalars cx and V u,v in a vector space. Let us now look at some subsets of the underlying sets of vector spaces. 3.4 SUBSPACES In E3 yousaw that V, a subset of C, was also a vector space. You also saw, in Example 6, that the subset V = {(x,y) E R2Iy = 5x1, of the vector space R', is itself a vector space under the same operations as those in R' In these cases V is a subspace of R2. Let us see what this means.