Linear Algebra I: Vector Spaces A 1 Vector spaces and subspaces 1.1 Let F be a field (in this book, it will always be either the field of reals R or the field of complex numbers C). A vector space V D .V; C; o;˛./.˛2 F// over F is a set V with a binary operation C, a constant o and a collection of unary operations (i.e. maps) ˛ W V ! V labelled by the elements of F, satisfying (V1) .x C y/ C z D x C .y C z/, (V2) x C y D y C x, (V3) 0 x D o, (V4) ˛ .ˇ x/ D .˛ˇ/ x, (V5) 1 x D x, (V6) .˛ C ˇ/ x D ˛ x C ˇ x,and (V7) ˛ .x C y/ D ˛ x C ˛ y. Here, we write ˛ x and we will write also ˛x for the result ˛.x/ of the unary operation ˛ in x. Often, one uses the expression “multiplication of x by ˛”; but it is useful to keep in mind that what we really have is a collection of unary operations (see also 5.1 below). The elements of a vector space are often referred to as vectors. In contrast, the elements of the field F are then often referred to as scalars. In view of this, it is useful to reflect for a moment on the true meaning of the axioms (equalities) above. For instance, (V4), often referred to as the “associative law” in fact states that the composition of the functions V ! V labelled by ˇ; ˛ is labelled by the product ˛ˇ in F, the “distributive law” (V6) states that the (pointwise) sum of the mappings labelled by ˛ and ˇ is labelled by the sum ˛ C ˇ in F, and (V7) states that each of the maps ˛ preserves the sum C. See Example 3 in 1.2. I. Kriz and A. Pultr, Introduction to Mathematical Analysis, 451 DOI 10.1007/978-3-0348-0636-7, © Springer Basel 2013 452 A Linear Algebra I: Vector Spaces 1.2 Examples Vector spaces are ubiquitous. We present just a few examples; the reader will certainly be able to think of many more. 1. The n-dimensional row vector space Fn. The elements of Fn are the n-tuples .x1;:::;xn/ with xi 2 F, the addition is given by .x1;:::;xn/ C .y1;:::;yn/ D .x1 C y1;:::;xn C yn/; o D .0;:::;0/,andthe˛’s operate by the rule ˛..x1;:::;xn// D .˛x1;:::;˛xn/: Note that F1 can be viewed as the F. However, although the operations acome from the binary multiplication in F, their role in a vector space is different. See 5.1 below. 2. Spaces of real functions.ThesetF.M/ of all real functions on a set M , with pointwise addition and multiplication by real numbers is obviously a vector space over R. Similarly, we have the vector space C.J/ of all the continuous functions on an interval J , or e.g. the space C 1.J / of all continuously differentiable functions on an open interval J or the space C 1.J / of all smooth functions on J , i.e. functions which have all higher derivatives. There are also analogous C-vector spaces of complex functions. 3. Let V be the set of positive reals. Define x ˚ y D xy, o D 1, and for arbitrary ˛ 2 R, ˛ x D x˛.Then.V; ˚; o;˛ ./.˛ 2 R// is a vector space (see Exercise (1)). 1.3 An important convention We have distinguished above the elements of the vector space and the elements of the field by using roman and greek letters. This is a good convention for a definition, but in the row vector spaces Fn, which will play a particular role below, it is somemewhat clumsy. Instead, we will use for an arithmetic vector a bold-faced variant of the letter denoting the coordinates. Thus, x D .x1;:::;xn/; a D .a1;:::;an/; etc. Similarly we will write f D .f1;:::;fn/ for the n-tuple of functions fj W X ! R resp. C (after all, they can be viewed as mappings f W X ! Fn), and similarly. 1 Vector spaces and subspaces 453 These conventions make reading about vectors much easier, and we will maintain them as long as possible (for example in our discussion of multivariable differential calculus in Chapter 3). The fact is, however, that in certain more advanced settings the conventions become cumbersome or even ambiguous (for example in the context of tensor calculus in Chapter 15), and because of this, in the later chapters of this book we eventually abandon them, as one usually does in more advanced topics of analysis. We do, however, use the symbol o universally for the zero element of a general vector space – so that in Fn we have o D .0;0;:::;0/. 1.4 We have the following trivial Observation. In any vector space V , for all x 2 V , we have x C o D x and there exists precisely one y such that x C y D o, namely y D .1/x. (Indeed, x C o D 1 x C 0 x D .1 C 0/x D x and x C .1/x D 1x C .1/x D .1 C .1//x D 0 x D o,andifx C y D o and x C z D o then y D y C .x C z/ D .y C x/ C z D z.) 1.5 (Vector) subspaces A subspace of a vector space V is a subset W  V that is itself a vector space with the operations inherited from V . Since the equations required in V hold for special as well as general elements, we have a trivial Observation. A subset W  V of a vector space is a subspace if and only if (a) o 2 W , (b) if x;y 2 W then x C y 2 W , and (c) for all ˛ 2 F and x 2 W , ˛x 2 W . 1.5.1 Also the following statement is immediate. Proposition. The intersection of an arbitrary set of subspaces of a vector space V is a subspace of V . 1.6 Generating sets By 1.5.1, we see that for each subset M of V there exists the smallest subspace W  V containing M , namely 454 A Linear Algebra I: Vector Spaces \ L.M / D fW j W subspace of V and M  W g: For M finite, we use the notatiom L.u1;:::;un/ instead of L.fu1;:::;ung/: Obviously L.;/ Dfog. We say that M generates L.M /; in particular if L.M / D V we say that M is a generating set (of V ). One often speaks of a set of generators but we have to keep in mind that this does not imply each of its elements generates V , which would be a much stronger statement. If there exists a finite generating system we say that V is finitely generated,or finite-dimensional. 1.7 The sum of subspaces Let W1;W2 be subspaces. Unlike the intersection W1 \ W2, the union W1 [ W2 is generally (and typically) not a subspace. But we have the smallest subspace containing both W1 and W2, namely L.W1 [ W2/. It will be denoted by W1 C W2 and called the sum of W1 and W2. (One often uses the symbol ‘˚’ instead of ‘C’ when one also has W1 \ W2 Dfog.) 2 Linear combinations, linear independence 2.1 A linear combination of a system x1;:::;xn of elements of a vector space V over F is a formula Xn ˛1x1 CC˛nxn (briefly, ˛j xj /: (*) j D1 The “system” in question is to be understood as the sequence, although the order in which it is presented will play no role. However, a possible repetition of an individual element is essential. Note that we spoke of (*) as of a “formula”. That is, we had in mind the full information involved (more pedantically, we could speak of the linear combination as of the sequence together with the mapping f1;:::;ng!F sending j to ˛j ). The vector obtained as the result of the indicated operations should be referred to as the result of the linear combination (*). We will follow this convention consistently 2 Linear combinations, linear independence 455 Xn to begin with; later, we will speak of a linear combination ˛j xj more loosely, j D1 trusting that the reader will be able to tell from the context whether we will mean the explicit formula or its result. 2.2 A linear combination (*)issaidtobenon-trivial if at least one of the ˛j is non-zero. Asystemx1;:::;xn is linearly dependent if there exists a non-trivial linear combination (*) with result o. Otherwise, we speak of a linearly independent system. 2.2.1 Proposition. 1. If x1;:::;xn is linearly dependent resp. independent then for any permutation of f1;:::;ng the system x.1/;:::;x.n/ is linearly dependent resp. independent. 2. A subsystem of a linearly independent system is linearly independent. 3. Let ˇ2;:::;ˇn be arbitrary. Then x1 :::;xn is linearly independent if and only if Xn the system x1 C ˇj xj ;x2;:::;xn is. j D2 4. A system x1;:::;xn is linearly dependent if and only if some of its members are a (result of a) linear combination of the others. In particular, any system containing o is linearly dependent. Similarly, if there exist j ¤ k such that xj D xk then x1;:::;xn is linearly dependent.