Linear Algebra

Chapter 1

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 linear algebra, better theorems and more insight emerge if complex numbers are investigated along with real numbers. Thus we will begin by introducing the complex numbers and their basic properties. We will generalize the examples of a plane and of ordinary space to 𝐑 𝑛 and 𝐂 𝑛, which we then will generalize to the notion of a vector space. As we will see, a vector space is a set with operations of addition and scalar multiplication that satisfy natural algebraic properties. Then our next topic will be subspaces, which play a role for vector spaces analogous to the role played by subsets for sets. Finally, we will look at sums of subspaces (analogous to unions of subsets) and direct sums of subspaces (analogous to unions of disjoint sets). Pierre Louis Dumesnil, Nils Forsberg René Descartes explaining his work to Queen Christina of Sweden. Vector spaces are a generalization of the description of a plane using two coordinates, as published by Descartes in 1637. © Sheldon Axler 2024 S. Axler, Linear Algebra Done Right, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-031-41026-0_1 1 

Chapter 1

2 1A Vector Spaces 𝐑 𝑛 and 𝐂 𝑛 Complex Numbers You should already be familiar with basic properties of the set 𝐑 of real numbers. Complex numbers were invented so that we can take square roots of negative numbers. The idea is to assume we have a square root of −1, denoted by 𝑖, that obeys the usual rules of arithmetic. Here are the formal definitions. 1.1 definition: complex numbers, 𝐂 • A complex number is an ordered pair (𝑎, 𝑏), where 𝑎, 𝑏 ∈ 𝐑 , but we will write this as 𝑎 + 𝑏𝑖. • The set of all complex numbers is denoted by 𝐂 : 𝐂 = {𝑎 + 𝑏𝑖 ∶ 𝑎, 𝑏 ∈ 𝐑}. • 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 . If 𝑎 ∈ 𝐑 , we identify 𝑎 + 0𝑖 with the real number 𝑎. Thus we think of 𝐑 as a subset of 𝐂 . We usually write 0 + 𝑏𝑖 as just 𝑏𝑖, and we usually write 0 + 1𝑖 as just 𝑖. To motivate the definition of complex The symbol 𝑖 was first used to denote multiplication given above, pretend that √−1 by Leonhard Euler in 1777. we knew that 𝑖2 = −1 and then use the usual rules of arithmetic to derive the formula above for the product of two complex numbers. Then use that formula to verify that we indeed have 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 (as given by 1.3). The next example illustrates this procedure. 1.2 example: complex arithmetic The product (2 + 3𝑖)(4 + 5𝑖) can be evaluated by applying the distributive and commutative properties from 1.3: (2 + 3𝑖)(4 + 5𝑖) = 2 ⋅ (4 + 5𝑖) + (3𝑖)(4 + 5𝑖) = 2 ⋅ 4 + 2 ⋅ 5𝑖 + 3𝑖 ⋅ 4 + (3𝑖)(5𝑖) = 8 + 10𝑖 + 12𝑖 − 15 = −7 + 22𝑖. 

Section 1A

𝐑 𝑛 and 𝐂 𝑛 3 Our first result states that complex addition and complex multiplication have the familiar properties that we expect. 1.3 properties of complex arithmetic commutativity 𝛼 + 𝛽 = 𝛽 + 𝛼 and 𝛼𝛽 = 𝛽𝛼 for all 𝛼, 𝛽 ∈ 𝐂 . associativity (𝛼 + 𝛽) + 𝜆 = 𝛼 + (𝛽 + 𝜆) and (𝛼𝛽)𝜆 = 𝛼(𝛽𝜆) for all 𝛼, 𝛽, 𝜆 ∈ 𝐂 . identities 𝜆 + 0 = 𝜆 and 𝜆1 = 𝜆 for all 𝜆 ∈ 𝐂 . additive inverse For every 𝛼 ∈ 𝐂 , there exists a unique 𝛽 ∈ 𝐂 such that 𝛼 + 𝛽 = 0. multiplicative inverse For every 𝛼 ∈ 𝐂 with 𝛼 ≠ 0, there exists a unique 𝛽 ∈ 𝐂 such that 𝛼𝛽 = 1. distributive property 𝜆(𝛼 + 𝛽) = 𝜆𝛼 + 𝜆𝛽 for all 𝜆, 𝛼, 𝛽 ∈ 𝐂 . The properties above are proved using the familiar properties of real numbers and the definitions of complex addition and multiplication. The next example shows how commutativity of complex multiplication is proved. Proofs of the other properties above are left as exercises. 1.4 example: commutativity of complex multiplication To show that 𝛼𝛽 = 𝛽𝛼 for all 𝛼, 𝛽 ∈ 𝐂 , suppose 𝛼 = 𝑎 + 𝑏𝑖 and 𝛽 = 𝑐 + 𝑑𝑖, where 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝐑 . Then the definition of multiplication of complex numbers shows that 𝛼𝛽 = (𝑎 + 𝑏𝑖)(𝑐 + 𝑑𝑖) = (𝑎𝑐 − 𝑏𝑑) + (𝑎𝑑 + 𝑏𝑐)𝑖 and 𝛽𝛼 = (𝑐 + 𝑑𝑖)(𝑎 + 𝑏𝑖) = (𝑐𝑎 − 𝑑𝑏) + (𝑐𝑏 + 𝑑𝑎)𝑖. The equations above and the commutativity of multiplication and addition of real numbers show that 𝛼𝛽 = 𝛽𝛼. 

4

Chapter 1 Vector Spaces Next, we define the additive and multiplicative inverses of complex numbers, and then use those inverses to define subtraction and division operations with complex numbers. 1.5 definition: −𝛼, subtraction, 1/𝛼, division Suppose 𝛼, 𝛽 ∈ 𝐂 . • Let −𝛼 denote the additive inverse of 𝛼. Thus −𝛼 is the unique complex number such that 𝛼 + (−𝛼) = 0. • Subtraction on 𝐂 is defined by 𝛽 − 𝛼 = 𝛽 + (−𝛼). • For 𝛼 ≠ 0, let 1/𝛼 and denote the multiplicative inverse of 𝛼. Thus 1/𝛼 is the unique complex number such that 1 𝛼 𝛼(1/𝛼) = 1. • For 𝛼 ≠ 0, division by 𝛼 is defined by 𝛽/𝛼 = 𝛽(1/𝛼). So that we can conveniently make definitions and prove theorems that apply to both real and complex numbers, we adopt the following notation. 1.6 notation: 𝐅 Throughout this book, 𝐅 stands for either 𝐑 or 𝐂 . Thus if we prove a theorem involving The letter 𝐅 is used because 𝐑 and 𝐂 𝐅, we will know that it holds when 𝐅 is are examples of what are called fields. replaced with 𝐑 and when 𝐅 is replaced with 𝐂 . Elements of 𝐅 are called scalars. The word “scalar” (which is just a fancy word for “number”) is often used when we want to emphasize that an object is a number, as opposed to a vector (vectors will be defined soon). For 𝛼 ∈ 𝐅 and 𝑚 a positive integer, we define 𝛼 𝑚 to denote the product of 𝛼 with itself 𝑚 times: 𝛼 𝑚 = ⏟. 𝛼⋯𝛼 𝑚 times This definition implies that 𝑛 (𝛼 𝑚 ) = 𝛼 𝑚𝑛 and (𝛼𝛽) 𝑚 = 𝛼 𝑚 𝛽 𝑚 for all 𝛼, 𝛽 ∈ 𝐅 and all positive integers 𝑚, 𝑛. 

Section 1A

𝐑 𝑛 and 𝐂 𝑛 5 Lists Before defining 𝐑 𝑛 and 𝐂 𝑛, we look at two important examples. 1.7 example: 𝐑2 and 𝐑3 • The set 𝐑2, which you can think of as a plane, is the set of all ordered pairs of real numbers: 𝐑2 = {(𝑥, 𝑦) ∶ 𝑥, 𝑦 ∈ 𝐑}. • The set 𝐑3, which you can think of as ordinary space, is the set 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. 1.8 definition: list, length • Suppose 𝑛 is a nonnegative integer. A list of length 𝑛 is an ordered collection of 𝑛 elements (which might be numbers, other lists, or more abstract objects). • Two lists are equal if and only if they have the same length and the same elements in the same order. Lists are often written as elements Many mathematicians call a list of separated by commas and surrounded by length 𝑛 an 𝑛-tuple. parentheses. Thus a list of length two is an ordered pair that might be written as (𝑎, 𝑏). A list of length three is an ordered triple that might be written as (𝑥, 𝑦, 𝑧). A list of length 𝑛 might look like this: (𝑧1 , …, 𝑧 𝑛 ). 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. Thus 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. Lists differ from sets in two ways: in lists, order matters and repetitions have meaning; in sets, order and repetitions are irrelevant. 1.9 example: lists versus sets • 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 do not have the same length), although the sets {4, 4} and {4, 4, 4} both equal the set {4}. 

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