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Chapter 1: Linearity: Basic Concepts and Examples

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Chapter 1: Linearity: Basic Concepts and Examples CHAPTER I LINEARITY: BASIC CONCEPTS AND EXAMPLES In this chapter we start with the concept of general linear spaces with elements in it called vectors, for “setting up the stage”. Then we introduce “actors” called linear mappings, which act upon vectors. In the mathematical literature, “vector spaces” is synonymous to “linear spaces” and these words will be used exchangeably. Also, “linear transformations” and “linear mappings” or simply “linear maps”, are also synonymous. 1. Linear Spaces and Linear Maps § 1.1. A vector space is an entity containing objects called vectors. A vector is usually conceived to be something which has a magnitude and direction, so that it can be drawn as an arrow: You can add or subtract two vectors: You can also multiply vectors by scalars: Such things should be familiar to you. However, we should not be so narrow-minded to think that only those objects repre- 1 sented geometrically by arrows in a 2D or 3D space can be regarded as vectors. As long as we have a collection of objects among which two algebraic operations called addition and scalar multiplication can be performed, so that certain rules of such operations are obeyed, we may regard this collection as a vector space and call the objects in this col- lection vectors. Our definition of vector spaces should be so general that we encounter vector spaces almost everyday and almost everywhere. Examples of vector spaces include many spaces of functions, spaces of polynomials, spaces of sequences etc. (in addition to the well-known 3D space in which vectors are represented as arrows.) The universality and the omnipresence of vector spaces is one good reason for placing linear algebra in the position of paramount importance in basic mathematics. 1.2. After this propaganda, we come to the technical side of the definition. We start with a collection V of objects called vectors, designated by block letters u, v, etc. Suppose that V is equipped with two algebraic operations: 1. Addition, allowing us to add two vectors u and v to obtain their sum u + v. 2. Scalar multiplication, allowing us to multiply a vector v by a scalar a to form av. Then V will be legitimately called a vector space if they obey a set of “natural” axioms. The complete set of axioms will be listed in Appendix A at the end of the present chapter. There is no need to memorize them, but here we mention some to show that they are indeed very natural. u + v = v + u (addition is commutative) u + (v + w) = (u + v) + w (addition is associative) u + 0 = u a(u + v) = au + av We are a bit vague about “scalars” in the above definition. What are scalars? The obvious answer is: they are numbers. But what sort of numbers are they? If we allow scalars to be complex numbers, then we say that V is a complex vector space, or a vector space over (the complex field) C. If we restrict scalars to real numbers, then V is called a real vector space, or a vector space over (the real field) R. More generally, scalars are taken from something called a field. If the “field of scalars” is denoted by F, we call V a vector space over F. In recent years, finite fields become an important subject due to its vast applications such as cryptography and coding theory. We only briefly describe about fields other than R and C in Appendix B at the end of the present chapter. In this course we only consider vector spaces over R or C. 2 1.3. Examples. The best way to understand the concept of vector spaces is to go through a large number of examples and work on them in the future. Example 1.3.1. Rn, a real vector space. The vectors in Rn are n-tuples of real numbers. A typical vector may be written as v = (v1, v2, . , vn), where v1, v2, . , vn are real numbers. Two n-tuples are equal only when their correspond- n ing components are equal: for u = (u1, . , un) and v = (v1, . , vn) in R , u = v only when u1 = v1, u2 = v2, . , un = vn. The addition and scalar multiplication in this space are defined in the componentwise manner: for u = (u1, u2 . , un), v = (v1, v2, . , vn) in Rn and a in R (that is, a is a real number), u + v = (u1 + v1, u2 + v2, . , un + vn), au = (au1, au2, . , aun). These two algebraic operations are quite simple and natural. Notice that, when n = 1, we have the vector space R1, which can be identified with R. This shows that R itself can be considered as a real vector space. Example 1.3.2. Cn, a complex vector space. The space Cn consists of all n-tuples of complex numbers. Everything in this space does in the same way as the space Rn of the previous example: all we have to do is replace real scalars by complex numbers. Sometimes we would like to make a statement for both spaces Rn and Cn. To avoid repetition, we use the letter F for R or C, or even a general field of scalars. We write Fn for the space of all n–tuples of scalars in F. Using the identification of F1 with F, we see that scalars can be regarded as vectors, if we wish. Example 1.3.3. Mmn(F)(the space of m n matrices over F.) × A vector in Fn is formed by arranging n scalars in a row. A variant of Fn is to form a vector by arranging mn scalars in a m n matrix. Denote by Mmn(F) the set of all m n × × matrices with entries in F. With the usual addition and scalar multiplication, Mmn(F) is a vector space. A vector in this space is actually a matrix. In the future, to simplify our notation, we will write Mmn for Mmn(F), if which field F we are working with is understood. Example 1.3.4. Direct product. This example tells us how to construct a new vector space from given vector spaces. Let V1,V2,...,Vn be vector spaces over the same field F. Consider the set V of all n-tuples 3 of the form v = (v1, v2,..., vn) where v1 is a vector in V1, v2 as a vector in V2, etc. The addition and the scalar multiplication for V is defined in the same fashion as the n corresponding operations of R described in Example 1.3.1: for u = (u1, u2,..., un) and v = (v , v ,..., vn) in V , and for a F, 1 2 ∈ u + v = (u1 + v1, u2 + v2,......, un + vn) au = (au1, au2, . , aun). This space V is a generalization of the previous example of Fn by replacing numbers in n the entries of n-tuples by vectors. If we take V = V = = Vn = F, then we recover F . 1 2 ··· The vector space V constructed in this way is called the direct product of V1,V2 ...,Vn n and the usual notation to express this is V = V V Vn, or V = Π Vk. 1 × 2 × · · · × k= 1 Example 1.3.5. Space of functions. This example is a space of functions with a common domain, say X (some nonempty set). A function f on X is just a way to assign to each point x in X a value denoted by f(x). This value could be a scalar or a vector. Since scalars can be regarded as vectors, so it is enough to consider the vector–valued case. Take any vector space V with F as its field of scalars. Denote by (X, V ) the set of all functions f from X to V : f assigns to each F point x in X to a vector denoted by f(x) in V . Given “vectors” (which are functions in this case) f and g in (X, V ), the sum f + g and the scalar multiple af of f a scalar a F are formally defined as follows (f + g)(x) = f(x) + g(x), (af)(x) = a.f(x), x X. ∈ If we treat x as a variable symbol so that f is written as f(x), then the sum of “vectors” f(x) and g(x) in (X, V ) is simply f(x) + g(x). In the special case with V = C and F X = N = 0, 1, 2, 3,... , { } each f (N, C) represents a sequence (of complex numbers): ∈ F f(n) n≥ (f(0), f(1), f(2), ) { } 0 ≡ ··· Conversely, each sequence an n≥ of complex numbers determines a function f on N, { } 0 given by f(n) = an. Thus we can identify the space (N, C) with the space of sequences, F which will be denoted by . For a = an n≥ and b = bn n≥ in , we define the S { } 0 { } 0 S sum a + b and the scalar multiple λa of a by a complex number λ as follows: a + b = (a0, a1, a2,... ) + (b0, b1, b2,... ) = (a0 + b0, a1 + b1, a2 + b2,... ) λa = λ(a0, a1, a2,... ) = (λa0, λa1, λa2,... ) 4 We will need the sequence space to study difference equations and recursion relations. S Example 1.3.6. The space of all polynomials. We specialize the previous example by taking X = C, the complex plane, and V = C, considered as a complex vector space. In some sense, the space (C, C) is too big. We F should look at something smaller inside. Recall that a polynomial is a function p on R which can be written in the form 2 3 n p(x) = a + a x + a x + a x + + anx , 0 1 2 3 ··· where a0, a1 .
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