Vector Spaces

Chapter 1 Vector Spaces Linear algebra is the study of linear maps on finite-dimensional vector spaces. Eventually we will learn what all these terms mean. In this chapter we will define vector spaces and discuss their elementary properties. In some areas of mathematics, including linear algebra, better theorems and more insight emerge if complex numbers are investigated along with real numbers. Thus we begin by introducing the complex numbers and their basic✽ properties. 1 2 Chapter 1. Vector Spaces Complex Numbers You should already be familiar with the basic properties of the set R of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The key idea is to assume we have i − i The symbol was√ first a square root of 1, denoted , and manipulate it using the usual rules used to denote −1 by of arithmetic. Formally, a complex number is an ordered pair (a, b), the Swiss where a, b ∈ R, but we will write this as a + bi. The set of all complex mathematician numbers is denoted by C: Leonhard Euler in 1777. C ={a + bi : a, b ∈ R}. If a ∈ R, we identify a + 0i with the real number a. Thus we can think of R as a subset of C. Addition and multiplication on C are defined by (a + bi) + (c + di) = (a + c) + (b + d)i, (a + bi)(c + di) = (ac − bd) + (ad + bc)i; here a, b, c, d ∈ R. Using multiplication as defined above, you should verify that i2 =−1. Do not memorize the formula for the product of two complex numbers; you can always rederive it by recalling that i2 =−1 and then using the usual rules of arithmetic. You should verify, using the familiar properties of the real numbers, that addition and multiplication on C satisfy the following properties: commutativity w + z = z + w and wz = zw for all w,z ∈ C; associativity (z1 + z2) + z3 = z1 + (z2 + z3) and (z1z2)z3 = z1(z2z3) for all z1,z2,z3 ∈ C; identities z + 0 = z and z1 = z for all z ∈ C; additive inverse for every z ∈ C, there exists a unique w ∈ C such that z + w = 0; multiplicative inverse for every z ∈ C with z = 0, there exists a unique w ∈ C such that zw = 1; Complex Numbers 3 distributive property λ(w + z) = λw + λz for all λ, w, z ∈ C. For z ∈ C, we let −z denote the additive inverse of z. Thus −z is the unique complex number such that z + (−z) = 0. Subtraction on C is defined by w − z = w + (−z) for w,z ∈ C. For z ∈ C with z = 0, we let 1/z denote the multiplicative inverse of z. Thus 1/z is the unique complex number such that z(1/z) = 1. Division on C is defined by w/z = w(1/z) for w,z ∈ C with z = 0. So that we can conveniently make definitions and prove theorems that apply to both real and complex numbers, we adopt the following notation: Throughout this book, The letter F is used F stands for either R or C. because R and C are examples of what are Thus if we prove a theorem involving F, we will know that it holds when called fields. In this F is replaced with R and when F is replaced with C. Elements of F are book we will not need called scalars. The word “scalar”, which means number, is often used to deal with fields other when we want to emphasize that an object is a number, as opposed to than R or C. Many of a vector (vectors will be defined soon). the definitions, m For z ∈ F and m a positive integer, we define z to denote the theorems, and proofs product of z with itself m times: in linear algebra that work for both R and C zm = z ·····z . also work without m times change if an arbitrary m n mn m m m R C Clearly (z ) = z and (wz) = w z for all w,z ∈ F and all field replaces or . positive integers m, n. 4 Chapter 1. Vector Spaces Definition of Vector Space Before defining what a vector space is, let’s look at two important examples. The vector space R2, which you can think of as a plane, consists of all ordered pairs of real numbers: R2 ={(x, y) : x,y ∈ R}. The vector space R3, which you can think of as ordinary space, consists of all ordered triples of real numbers: R3 ={(x,y,z): x,y,z ∈ R}. To generalize R2 and R3 to higher dimensions, we first need to discuss the concept of lists. Suppose n is a nonnegative integer. A list of length n is an ordered collection of n objects (which might be numbers, other lists, or more abstract entities) separated by commas and Many mathematicians surrounded by parentheses. A list of length n looks like this: call a list of length n an (x ,...,x ). n-tuple. 1 n Thus a list of length 2 is an ordered pair and a list of length 3 is an th ordered triple. For j ∈{1,...,n}, we say that xj is the j coordinate of the list above. Thus x1 is called the first coordinate, x2 is called the second coordinate, and so on. Sometimes we will use the word list without specifying its length. Remember, however, that by definition each list has a finite length that is a nonnegative integer, so that an object that looks like (x1,x2,...), which might be said to have infinite length, is not a list. A list of length 0 looks like this: (). We consider such an object to be a list so that some of our theorems will not have trivial exceptions. Two lists are equal if and only if they have the same length and the same coordinates in the same order. In other words, (x1,...,xm) equals (y1,...,yn) if and only if m = n and x1 = y1,...,xm = ym. Lists differ from sets in two ways: in lists, order matters and repetitions are allowed, whereas in sets, order and repetitions are irrelevant. For example, the lists (3, 5) and (5, 3) are not equal, but the sets {3, 5} and {5, 3} are equal. The lists (4, 4) and (4, 4, 4) are not equal (they Definition of Vector Space 5 do not have the same length), though the sets {4, 4} and {4, 4, 4} both equal the set {4}. To define the higher-dimensional analogues of R2 and R3, we will simply replace R with F (which equals R or C) and replace the 2 or 3 with an arbitrary positive integer. Specifically, fix a positive integer n for the rest of this section. We define Fn to be the set of all lists of length n consisting of elements of F: n F ={(x1,...,xn) : xj ∈ F for j = 1,...,n}. For example, if F = R and n equals 2 or 3, then this definition of Fn agrees with our previous notions of R2 and R3. As another example, C4 is the set of all lists of four complex numbers: 4 C ={(z1,z2,z3,z4) : z1,z2,z3,z4 ∈ C}. If n ≥ 4, we cannot easily visualize Rn as a physical object. The same For an amusing problem arises if we work with complex numbers: C1 can be thought account of how R3 of as a plane, but for n ≥ 2, the human brain cannot provide geometric would be perceived by models of Cn. However, even if n is large, we can perform algebraic a creature living in R2, manipulations in Fn as easily as in R2 or R3. For example, addition is read Flatland: A defined on Fn by adding corresponding coordinates: Romance of Many Dimensions, by Edwin 1.1 (x1,...,xn) + (y1,...,yn) = (x1 + y1,...,xn + yn). A. Abbott. This novel, published in 1884, can n Often the mathematics of F becomes cleaner if we use a single help creatures living in entity to denote an list of n numbers, without explicitly writing the three-dimensional n coordinates. Thus the commutative property of addition on F should space, such as be expressed as ourselves, imagine a x + y = y + x physical space of four or more dimensions. for all x,y ∈ Fn, rather than the more cumbersome (x1,...,xn) + (y1,...,yn) = (y1,...,yn) + (x1,...,xn) for all x1,...,xn,y1,...,yn ∈ F (even though the latter formulation is needed to prove commutativity). If a single letter is used to denote an element of Fn, then the same letter, with appropriate subscripts, is often used when coordinates must be displayed. For example, if n x ∈ F , then letting x equal (x1,...,xn) is good notation. Even better, work with just x and avoid explicit coordinates, if possible. 6 Chapter 1. Vector Spaces We let 0 denote the list of length n all of whose coordinates are 0: 0 = (0,...,0). Note that we are using the symbol 0 in two different ways—on the left side of the equation above, 0 denotes a list of length n, whereas on the right side, each 0 denotes a number. This potentially confusing practice actually causes no problems because the context always makes clear what is intended. For example, consider the statement that 0 is an additive identity for Fn: x + 0 = x for all x ∈ Fn. Here 0 must be a list because we have not defined the sum of an element of Fn (namely, x) and the number 0.

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