DOCSLIB.ORG

Section 4.4 Functions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Section 4.4 Functions Section 4.4 Functions CS 130 – Discrete Structures Function Definitions • Let S and T be sets. A function f from S to T, f: S T, is a subset of S x T where each member of S appears exactly once as the first component of an ordered pair. – S is the domain of the function. – T is the codomain of the function. • If (s, t) belongs to the function, then t is denoted by f(s) – t is the image of s under f – s is a preimage of t under f – f is said to map s to t CS 130 – Discrete Structures 39 Diagrams Representation • If A and B are finite sets, an arrow diagram shows a function f from A to B by drawing an arrow from each element in A to the corresponding element of B • Two properties must be held in the arrow diagram according to the definition of function: – Every element of A has an arrow coming out of it – No one element of A has two arrows coming out of it that point to two different elements of B • Example CS 130 – Discrete Structures 40 Examples • Which of the following are functions? – f: S T where S = T = {1, 2, 3}, f = {(1,1),(2,3),(2,1)} – g: Z N where g(x) = |x| (absolute value of x) – h: N N where h(x) = x – 4 – f: R R where f(x) = 4x – 1 – more examples from the book • For f: Z Z where f(x) = x2 – what is the image of –4 – what are the preimages of 9 CS 130 – Discrete Structures 41 Several Common Math Functions • Floor function – associates with each real number x the greatest integer less than or equal to x – E.g., 2.8 = ?, - 2.8 = ? • Ceiling function – associates with each real number x the smallest integer greater than or equal to x – E.g, 2.8 = ?, -2.8 = ? • Modulo function f(x) = x mod n – associates with x the remainder when x is divided by n – we can write x = qn+ r, r is between [0, n-1] – 10 mod 3 = 1 CS 130 – Discrete Structures 42 Functions With Multiple Variables • A function can include more than one variable. A function can be defined as – f: S1 S2 …… Sn T that associates with each ordered n-tuple of elements (s1, s2, …, sn) – Example: f: Z Z Z is given by f(x,y) = x+y CS 130 – Discrete Structures 43 Properties of Functions: Surjective • Three properties: surjective (onto), injective, bijective • Let f: S T be an arbitrary function – every member of S has an image under f and all the images are members of T – the set R of all such images is called the range of the function f • A function f: S T is an onto, or surjective, function if the range of f equals the codomain of f CS 130 – Discrete Structures 44 Example of Surjective Functions • To prove a function to be surjective: need to show that an arbitrary member of the codomain T is a member of the range R, thus it is the image of some member of the domain, we have T R • To disprove it: if we can find one member of the codomain that is not the image of any member of the domain • Let f: Q Q where f(x) = 3x + 2 • Let g: Z N where g(x) = |x| • Let f: R R where f(x) = 4x – 1 CS 130 – Discrete Structures 45 Properties of Functions: Injective, Bijective • A function f: S T is an one-to-one or injective, if no member of T is the image under f of two distinct elements of S – To prove a function is injective: we assume that there are elements s1 and s2 of S with f(s1) = f(s2) and then show that s1 = s2 – To disprove it: counterexample, where an element in the range has two preimages in the domain • A function f: S T is bijective, if it is both surjective and injective CS 130 – Discrete Structures 46 Examples • The function g: R R where g(x) = x3 – surjective, injective, bijective • The function f: N N where f(x) = x2 – not surjective, injective, not bijective CS 130 – Discrete Structures 47 In General CS 130 – Discrete Structures 48 Composition of Functions • Let fS : T and gT : U . Then the composition function, g f , is a function from S to U defined by ( gfs )() gfs (()) . – Function f is applied first, and then function g – It is not always possible to take any two arbitrary functions and compose them since the domain and the ranges have to be compatible. – Note that composition preserves the properties of being onto and being one-to-one • Composition on two bijections is a bijection CS 130 – Discrete Structures 49 Examples • Let f: R R be defined by f(x) = x2 and g: R R be defined by g(x) = x – What is the value of (g f) (2.3) ? • g(f(2.3)) = g(5.39) = 5 – What is the value of (f g) (2.3) ? • f(g(2.3)) = f(2) = 4 – Order is important in function composition • The following functions map R to R. Give an equation describing the composition functions f g and g f in each case: – f(x) = 3x2, g(x) = 5x CS 130 – Discrete Structures 50 More Examples • Given the following function, decide whether it is 1-to-1 or onto: • f: NN, f(x) = x + 1 but 1-to-1 Proof that f is one-to-one. Let f(s1) = f(s2) for s1, s2 . Then s1 + 1 = s2 + 1 definition of f s1 = s2 algebra CS 130 – Discrete Structures f is one-to-one 51 Proofs • Let f: S T and g: T U, and assume that both f and g are one-to-one (injective) functions. Prove that g f is a one-to-one function. • Let f: S T and g: T U be functions – Prove that if g f is 1-to-1, so is f. – Prove that if g f is onto, so is g. – Find an example where g f is 1-to-1, but g is not 1- to-1. – Find an example where g f is onto but f is not onto. CS 130 – Discrete Structures 52 Inverse Functions • Bijective functions have another property: – Every element s in S has an image in T and every element of T has a unique inverse image in S since the f is onto and one-to-one • If there is a function f which has a one-to-one correspondence from a set S to a set T, then there is a function g from T to S that "undoes" the action of f. This function g is called the inverse function for f. • Then g f (s) = s maps each element to itself. Such a function which leave an element unchanged is called the identity function on S and is denoted by is • Show that f g = iT CS 130 – Discrete Structures 53 Inverse Functions • Definition: Let f: S T. If there exists a function g: T S, such that – g f = iS and f g = iT – Then g is called the inverse function of f and is denoted by f –1 • Example: f: R R given by f(x) = 3x + 4 is a bijection. Describe f –1 • Theorem on Bijections and Inverse Functions: – Let f: S T. Then f is a bijection if and only if f –1 exists CS 130 – Discrete Structures 54 More Proofs • Let f and g be bijections, f: ST and g: TU. Then f-1 and g-1 exist. Also g f is a bijection from S to U. CS 130 – Discrete Structures 55 Summary of Function Terminologies Term Meaning function A function f from set S to set T is a relationship between elements of S and elements of T where each element of S is related to a unique element of T. It is denoted by f: S T. domain Starting set for a function codomain Ending set for a function image Point that results from a mapping preimage Starting point for a mapping range Collection of all images of a domain onto (surjective) Range is the whole codomain; every codomain element has a preimage one-to-one (injective) No two elements in a domain map to the same place bijection One-to-one and onto identity function Maps each element of set to itself inverse function For a bijection, a new function that maps each codomain element back where it came from CS 130 – Discrete Structures 56 Exercises • Let S = {0, 2, 4, 6} and T = {1, 3, 5, 7}. Determine whether each of the following sets of ordered pairs is a function with domain S and codomain T. If so, it is 1-to-1? Is it onto? a. {(0, 2), (2, 4), (4, 6), (6, 0)} b. {(6, 3), (2, 1), (0, 3), (4, 5)} c. {(2, 3), (4, 7), (0, 1), (6, 5)} d. {(2, 1), (4, 5), (6, 3)} e. {(6, 1), (0, 3), (4, 1), (0, 7), (2, 5)} CS 130 – Discrete Structures 57 Exercises • Which of the following are functions from the domain to the codomain given? Which functions are 1-to-1 or onto or both? Describe the inverse function for any bijective function. a. f: ZN where f(x) = x2 + 1 b. g: NQ where g(x) = 1/x c. h: Z x N Q where h(z, n) = z/(n+1) d.
Load more

Recommended publications
  • A Proof of Cantor's Theorem
    Cantor’s Theorem Joe Roussos 1 Preliminary ideas Two sets have the same number of elements (are equinumerous, or have the same cardinality) iff there is a bijection between the two sets. Mappings: A mapping, or function, is a rule that associates elements of one set with elements of another set. We write this f : X ! Y , f is called the function/mapping, the set X is called the domain, and Y is called the codomain. We specify what the rule is by writing f(x) = y or f : x 7! y. e.g. X = f1; 2; 3g;Y = f2; 4; 6g, the map f(x) = 2x associates each element x 2 X with the element in Y that is double it. A bijection is a mapping that is injective and surjective.1 • Injective (one-to-one): A function is injective if it takes each element of the do- main onto at most one element of the codomain. It never maps more than one element in the domain onto the same element in the codomain. Formally, if f is a function between set X and set Y , then f is injective iff 8a; b 2 X; f(a) = f(b) ! a = b • Surjective (onto): A function is surjective if it maps something onto every element of the codomain. It can map more than one thing onto the same element in the codomain, but it needs to hit everything in the codomain. Formally, if f is a function between set X and set Y , then f is surjective iff 8y 2 Y; 9x 2 X; f(x) = y Figure 1: Injective map.
    [Show full text]
  • 1 Elementary Set Theory
    1 Elementary Set Theory Notation: fg enclose a set. f1; 2; 3g = f3; 2; 2; 1; 3g because a set is not defined by order or multiplicity. f0; 2; 4;:::g = fxjx is an even natural numberg because two ways of writing a set are equivalent. ; is the empty set. x 2 A denotes x is an element of A. N = f0; 1; 2;:::g are the natural numbers. Z = f:::; −2; −1; 0; 1; 2;:::g are the integers. m Q = f n jm; n 2 Z and n 6= 0g are the rational numbers. R are the real numbers. Axiom 1.1. Axiom of Extensionality Let A; B be sets. If (8x)x 2 A iff x 2 B then A = B. Definition 1.1 (Subset). Let A; B be sets. Then A is a subset of B, written A ⊆ B iff (8x) if x 2 A then x 2 B. Theorem 1.1. If A ⊆ B and B ⊆ A then A = B. Proof. Let x be arbitrary. Because A ⊆ B if x 2 A then x 2 B Because B ⊆ A if x 2 B then x 2 A Hence, x 2 A iff x 2 B, thus A = B. Definition 1.2 (Union). Let A; B be sets. The Union A [ B of A and B is defined by x 2 A [ B if x 2 A or x 2 B. Theorem 1.2. A [ (B [ C) = (A [ B) [ C Proof. Let x be arbitrary. x 2 A [ (B [ C) iff x 2 A or x 2 B [ C iff x 2 A or (x 2 B or x 2 C) iff x 2 A or x 2 B or x 2 C iff (x 2 A or x 2 B) or x 2 C iff x 2 A [ B or x 2 C iff x 2 (A [ B) [ C Definition 1.3 (Intersection).
    [Show full text]
  • Linear Transformation (Sections 1.8, 1.9) General View: Given an Input, the Transformation Produces an Output
    Linear Transformation (Sections 1.8, 1.9) General view: Given an input, the transformation produces an output. In this sense, a function is also a transformation. 1 4 3 1 3 Example. Let A = and x = 1 . Describe matrix-vector multiplication Ax 2 0 5 1 1 1 in the language of transformation. 1 4 3 1 31 5 Ax b 2 0 5 11 8 1 Vector x is transformed into vector b by left matrix multiplication Definition and terminologies. Transformation (or function or mapping) T from Rn to Rm is a rule that assigns to each vector x in Rn a vector T(x) in Rm. • Notation: T: Rn → Rm • Rn is the domain of T • Rm is the codomain of T • T(x) is the image of vector x • The set of all images T(x) is the range of T • When T(x) = Ax, A is a m×n size matrix. Range of T = Span{ column vectors of A} (HW1.8.7) See class notes for other examples. Linear Transformation --- Existence and Uniqueness Questions (Section 1.9) Definition 1: T: Rn → Rm is onto if each b in Rm is the image of at least one x in Rn. • i.e. codomain Rm = range of T • When solve T(x) = b for x (or Ax=b, A is the standard matrix), there exists at least one solution (Existence question). Definition 2: T: Rn → Rm is one-to-one if each b in Rm is the image of at most one x in Rn. • i.e. When solve T(x) = b for x (or Ax=b, A is the standard matrix), there exists either a unique solution or none at all (Uniqueness question).
    [Show full text]
  • Inverse of Exponential Functions Are Logarithmic Functions
    Math Instructional Framework Full Name Math III Unit 3 Lesson 2 Time Frame Unit Name Logarithmic Functions as Inverses of Exponential Functions Learning Task/Topics/ Task 2: How long Does It Take? Themes Task 3: The Population of Exponentia Task 4: Modeling Natural Phenomena on Earth Culminating Task: Traveling to Exponentia Standards and Elements MM3A2. Students will explore logarithmic functions as inverses of exponential functions. c. Define logarithmic functions as inverses of exponential functions. Lesson Essential Questions How can you graph the inverse of an exponential function? Activator PROBLEM 2.Task 3: The Population of Exponentia (Problem 1 could be completed prior) Work Session Inverse of Exponential Functions are Logarithmic Functions A Graph the inverse of exponential functions. B Graph logarithmic functions. See Notes Below. VOCABULARY Asymptote: A line or curve that describes the end behavior of the graph. A graph never crosses a vertical asymptote but it may cross a horizontal or oblique asymptote. Common logarithm: A logarithm with a base of 10. A common logarithm is the power, a, such that 10a = b. The common logarithm of x is written log x. For example, log 100 = 2 because 102 = 100. Exponential functions: A function of the form y = a·bx where a > 0 and either 0 < b < 1 or b > 1. Logarithmic functions: A function of the form y = logbx, with b 1 and b and x both positive. A logarithmic x function is the inverse of an exponential function. The inverse of y = b is y = logbx. Logarithm: The logarithm base b of a number x, logbx, is the power to which b must be raised to equal x.
    [Show full text]
  • Analysis of Functions of a Single Variable a Detailed Development
    ANALYSIS OF FUNCTIONS OF A SINGLE VARIABLE A DETAILED DEVELOPMENT LAWRENCE W. BAGGETT University of Colorado OCTOBER 29, 2006 2 For Christy My Light i PREFACE I have written this book primarily for serious and talented mathematics scholars , seniors or first-year graduate students, who by the time they finish their schooling should have had the opportunity to study in some detail the great discoveries of our subject. What did we know and how and when did we know it? I hope this book is useful toward that goal, especially when it comes to the great achievements of that part of mathematics known as analysis. I have tried to write a complete and thorough account of the elementary theories of functions of a single real variable and functions of a single complex variable. Separating these two subjects does not at all jive with their development historically, and to me it seems unnecessary and potentially confusing to do so. On the other hand, functions of several variables seems to me to be a very different kettle of fish, so I have decided to limit this book by concentrating on one variable at a time. Everyone is taught (told) in school that the area of a circle is given by the formula A = πr2: We are also told that the product of two negatives is a positive, that you cant trisect an angle, and that the square root of 2 is irrational. Students of natural sciences learn that eiπ = 1 and that sin2 + cos2 = 1: More sophisticated students are taught the Fundamental− Theorem of calculus and the Fundamental Theorem of Algebra.
    [Show full text]
  • An Update on the Four-Color Theorem Robin Thomas
    thomas.qxp 6/11/98 4:10 PM Page 848 An Update on the Four-Color Theorem Robin Thomas very planar map of connected countries the five-color theorem (Theorem 2 below) and can be colored using four colors in such discovered what became known as Kempe chains, a way that countries with a common and Tait found an equivalent formulation of the boundary segment (not just a point) re- Four-Color Theorem in terms of edge 3-coloring, ceive different colors. It is amazing that stated here as Theorem 3. Esuch a simply stated result resisted proof for one The next major contribution came in 1913 from and a quarter centuries, and even today it is not G. D. Birkhoff, whose work allowed Franklin to yet fully understood. In this article I concentrate prove in 1922 that the four-color conjecture is on recent developments: equivalent formulations, true for maps with at most twenty-five regions. The a new proof, and progress on some generalizations. same method was used by other mathematicians to make progress on the four-color problem. Im- Brief History portant here is the work by Heesch, who developed The Four-Color Problem dates back to 1852 when the two main ingredients needed for the ultimate Francis Guthrie, while trying to color the map of proof—“reducibility” and “discharging”. While the the counties of England, noticed that four colors concept of reducibility was studied by other re- sufficed. He asked his brother Frederick if it was searchers as well, the idea of discharging, crucial true that any map can be colored using four col- for the unavoidability part of the proof, is due to ors in such a way that adjacent regions (i.e., those Heesch, and he also conjectured that a suitable de- sharing a common boundary segment, not just a velopment of this method would solve the Four- point) receive different colors.
    [Show full text]
  • Self-Organizing Tuple Reconstruction in Column-Stores
    Self-organizing Tuple Reconstruction in Column-stores Stratos Idreos Martin L. Kersten Stefan Manegold CWI Amsterdam CWI Amsterdam CWI Amsterdam The Netherlands The Netherlands The Netherlands [email protected] [email protected] [email protected] ABSTRACT 1. INTRODUCTION Column-stores gained popularity as a promising physical de- A prime feature of column-stores is to provide improved sign alternative. Each attribute of a relation is physically performance over row-stores in the case that workloads re- stored as a separate column allowing queries to load only quire only a few attributes of wide tables at a time. Each the required attributes. The overhead incurred is on-the-fly relation R is physically stored as a set of columns; one col- tuple reconstruction for multi-attribute queries. Each tu- umn for each attribute of R. This way, a query needs to load ple reconstruction is a join of two columns based on tuple only the required attributes from each relevant relation. IDs, making it a significant cost component. The ultimate This happens at the expense of requiring explicit (partial) physical design is to have multiple presorted copies of each tuple reconstruction in case multiple attributes are required. base table such that tuples are already appropriately orga- Each tuple reconstruction is a join between two columns nized in multiple different orders across the various columns. based on tuple IDs/positions and becomes a significant cost This requires the ability to predict the workload, idle time component in column-stores especially for multi-attribute to prepare, and infrequent updates. queries [2, 6, 10].
    [Show full text]
  • The Matrix Calculus You Need for Deep Learning
    The Matrix Calculus You Need For Deep Learning Terence Parr and Jeremy Howard July 3, 2018 (We teach in University of San Francisco's MS in Data Science program and have other nefarious projects underway. You might know Terence as the creator of the ANTLR parser generator. For more material, see Jeremy's fast.ai courses and University of San Francisco's Data Institute in- person version of the deep learning course.) HTML version (The PDF and HTML were generated from markup using bookish) Abstract This paper is an attempt to explain all the matrix calculus you need in order to understand the training of deep neural networks. We assume no math knowledge beyond what you learned in calculus 1, and provide links to help you refresh the necessary math where needed. Note that you do not need to understand this material before you start learning to train and use deep learning in practice; rather, this material is for those who are already familiar with the basics of neural networks, and wish to deepen their understanding of the underlying math. Don't worry if you get stuck at some point along the way|just go back and reread the previous section, and try writing down and working through some examples. And if you're still stuck, we're happy to answer your questions in the Theory category at forums.fast.ai. Note: There is a reference section at the end of the paper summarizing all the key matrix calculus rules and terminology discussed here. arXiv:1802.01528v3 [cs.LG] 2 Jul 2018 1 Contents 1 Introduction 3 2 Review: Scalar derivative rules4 3 Introduction to vector calculus and partial derivatives5 4 Matrix calculus 6 4.1 Generalization of the Jacobian .
    [Show full text]
  • I0 and Rank-Into-Rank Axioms
    I0 and rank-into-rank axioms Vincenzo Dimonte∗ July 11, 2017 Abstract Just a survey on I0. Keywords: Large cardinals; Axiom I0; rank-into-rank axioms; elemen- tary embeddings; relative constructibility; embedding lifting; singular car- dinals combinatorics. 2010 Mathematics Subject Classifications: 03E55 (03E05 03E35 03E45) 1 Informal Introduction to the Introduction Ok, that’s it. People who know me also know that it is years that I am ranting about a book about I0, how it is important to write it and publish it, etc. I realized that this particular moment is still right for me to write it, for two reasons: time is scarce, and probably there are still not enough results (or anyway not enough different lines of research). This has the potential of being a very nasty vicious circle: there are not enough results about I0 to write a book, but if nobody writes a book the diffusion of I0 is too limited, and so the number of results does not increase much. To avoid this deadlock, I decided to divulge a first draft of such a hypothetical book, hoping to start a “conversation” with that. It is literally a first draft, so it is full of errors and in a liquid state. For example, I still haven’t decided whether to call it I0 or I0, both notations are used in literature. I feel like it still lacks a vision of the future, a map on where the research could and should going about I0. Many proofs are old but never published, and therefore reconstructed by me, so maybe they are wrong.
    [Show full text]
  • IVC Factsheet Functions Comp Inverse
    Imperial Valley College Math Lab Functions: Composition and Inverse Functions FUNCTION COMPOSITION In order to perform a composition of functions, it is essential to be familiar with function notation. If you see something of the form “푓(푥) = [expression in terms of x]”, this means that whatever you see in the parentheses following f should be substituted for x in the expression. This can include numbers, variables, other expressions, and even other functions. EXAMPLE: 푓(푥) = 4푥2 − 13푥 푓(2) = 4 ∙ 22 − 13(2) 푓(−9) = 4(−9)2 − 13(−9) 푓(푎) = 4푎2 − 13푎 푓(푐3) = 4(푐3)2 − 13푐3 푓(ℎ + 5) = 4(ℎ + 5)2 − 13(ℎ + 5) Etc. A composition of functions occurs when one function is “plugged into” another function. The notation (푓 ○푔)(푥) is pronounced “푓 of 푔 of 푥”, and it literally means 푓(푔(푥)). In other words, you “plug” the 푔(푥) function into the 푓(푥) function. Similarly, (푔 ○푓)(푥) is pronounced “푔 of 푓 of 푥”, and it literally means 푔(푓(푥)). In other words, you “plug” the 푓(푥) function into the 푔(푥) function. WARNING: Be careful not to confuse (푓 ○푔)(푥) with (푓 ∙ 푔)(푥), which means 푓(푥) ∙ 푔(푥) . EXAMPLES: 푓(푥) = 4푥2 − 13푥 푔(푥) = 2푥 + 1 a. (푓 ○푔)(푥) = 푓(푔(푥)) = 4[푔(푥)]2 − 13 ∙ 푔(푥) = 4(2푥 + 1)2 − 13(2푥 + 1) = [푠푚푝푙푓푦] … = 16푥2 − 10푥 − 9 b. (푔 ○푓)(푥) = 푔(푓(푥)) = 2 ∙ 푓(푥) + 1 = 2(4푥2 − 13푥) + 1 = 8푥2 − 26푥 + 1 A function can even be “composed” with itself: c.
    [Show full text]
  • Determinacy in Linear Rational Expectations Models
    Journal of Mathematical Economics 40 (2004) 815–830 Determinacy in linear rational expectations models Stéphane Gauthier∗ CREST, Laboratoire de Macroéconomie (Timbre J-360), 15 bd Gabriel Péri, 92245 Malakoff Cedex, France Received 15 February 2002; received in revised form 5 June 2003; accepted 17 July 2003 Available online 21 January 2004 Abstract The purpose of this paper is to assess the relevance of rational expectations solutions to the class of linear univariate models where both the number of leads in expectations and the number of lags in predetermined variables are arbitrary. It recommends to rule out all the solutions that would fail to be locally unique, or equivalently, locally determinate. So far, this determinacy criterion has been applied to particular solutions, in general some steady state or periodic cycle. However solutions to linear models with rational expectations typically do not conform to such simple dynamic patterns but express instead the current state of the economic system as a linear difference equation of lagged states. The innovation of this paper is to apply the determinacy criterion to the sets of coefficients of these linear difference equations. Its main result shows that only one set of such coefficients, or the corresponding solution, is locally determinate. This solution is commonly referred to as the fundamental one in the literature. In particular, in the saddle point configuration, it coincides with the saddle stable (pure forward) equilibrium trajectory. © 2004 Published by Elsevier B.V. JEL classification: C32; E32 Keywords: Rational expectations; Selection; Determinacy; Saddle point property 1. Introduction The rational expectations hypothesis is commonly justified by the fact that individual forecasts are based on the relevant theory of the economic system.
    [Show full text]
  • SHEET 14: LINEAR ALGEBRA 14.1 Vector Spaces
    SHEET 14: LINEAR ALGEBRA Throughout this sheet, let F be a field. In examples, you need only consider the field F = R. 14.1 Vector spaces Definition 14.1. A vector space over F is a set V with two operations, V × V ! V :(x; y) 7! x + y (vector addition) and F × V ! V :(λ, x) 7! λx (scalar multiplication); that satisfy the following axioms. 1. Addition is commutative: x + y = y + x for all x; y 2 V . 2. Addition is associative: x + (y + z) = (x + y) + z for all x; y; z 2 V . 3. There is an additive identity 0 2 V satisfying x + 0 = x for all x 2 V . 4. For each x 2 V , there is an additive inverse −x 2 V satisfying x + (−x) = 0. 5. Scalar multiplication by 1 fixes vectors: 1x = x for all x 2 V . 6. Scalar multiplication is compatible with F :(λµ)x = λ(µx) for all λ, µ 2 F and x 2 V . 7. Scalar multiplication distributes over vector addition and over scalar addition: λ(x + y) = λx + λy and (λ + µ)x = λx + µx for all λ, µ 2 F and x; y 2 V . In this context, elements of F are called scalars and elements of V are called vectors. Definition 14.2. Let n be a nonnegative integer. The coordinate space F n = F × · · · × F is the set of all n-tuples of elements of F , conventionally regarded as column vectors. Addition and scalar multiplication are defined componentwise; that is, 2 3 2 3 2 3 2 3 x1 y1 x1 + y1 λx1 6x 7 6y 7 6x + y 7 6λx 7 6 27 6 27 6 2 2 7 6 27 if x = 6 .
    [Show full text]