DOCSLIB.ORG

Chapter 6, Ideals and Quotient Rings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 6, Ideals and Quotient Rings Chapter 6, Ideals and quotient rings Ideals. Finally we are ready to study kernels and images of ring homomorphisms. We have seen two major examples in which congruence gave us ring homomorphisms: Z → Zn and F [x] → F [x]=(p(x)). We shall generalize this to congruence in arbitrary rings and then see that it brings us very close to a complete understanding of kernels and images of ring homomorphisms. Recall the definition of a ring. For congruence, we need a special subring that will behave like nZ or like p(x)F [x]={p(x)f(x)|f(x)∈F[x]}. Definition, p. 135. A subring I of a ring R is an ideal if whenever r ∈ R and a ∈ I, then ra ∈ I and ar ∈ I. If R is commutative, we only need to worry about multiplication on one side. More generally, one can speak of left ideals and right ideals and two-sided ideals.Our main interest is in the two-sided ideals; these turn out to give us the congruences we want. Before we look at examples, recall that to be a subring means I is closed under multi- plication and subtraction. Thus we get Theorem 6.1. A nonempty subset I in a ring R is an ideal iff it satisfies (1) if a; b ∈ I,thena−b∈I; (2) if r ∈ R; a ∈ I,thenra ∈ I and ar ∈ I. Note that if R has an identity, we can replace (1) by (10) if a; b ∈ I,thena+b∈I because (2) implies that, since −1 ∈ R, −b ∈ I whenever b ∈ I. Examples. 1. nZ = { kn | k ∈ Z } for any n ∈ Z is an ideal in Z.Ifn=0wegetthe zero ideal, an ideal of any ring R.Ifn=1wegetZ, the whole ring. Again, for any ring R, the whole ring is an ideal of R. 2. In Z6, the set I = { [2k] ∈ Z6 | k ∈ Z } is an ideal. 3. p(x)R[x]={p(x)f(x)|f(x)∈R[x]}is an ideal of R[x] for any commutative ring R with 1. 4. In Z[x], the set I = { f(x) ∈ Z[x] | f(0) ≡ 0 (mod n) } is an ideal for any n ≥ 2inZ. This generalizes an example on page 136 where n =2. 5. For R = C(R; R), fix any r ∈ R. The set I = { f ∈ R | f(r)=0}is an ideal. Note that it does not work to use a number other than 0. 1 2 ab 6. For R = M2(R), the set of first rows I = a; b ∈ R is a right ideal but not 00 a left ideal. (A similar thing is done for columns and left ideals in the book.) In particular, I is not a (two-sided) ideal. Check. Examples 1, 2 and 3 above were all of a special type which we can generalize. Theorem 6.2. Let R be a commutative ring with identity. Let c ∈ R.Theset I={rc | r ∈ R } is an ideal of R. Proof. Given two elements r1c and r2c in I,wehaver1c−r2c=(r1−r2)c∈I. For any a ∈ R, a(r1c)=(ar1)c ∈ I. Therefore I is an ideal. (We have implicitly used the fact that R is commutative so that multiplication on the right also works.) We call the ideal in Theorem 6.2 the principal ideal generated by c and denote it by (c)orbyRc. The ideal in Example 4 was not principal. To see this, note that n ≥ 2 and x both lie in I.IfIwere generated by some polynomial p(x), then both n and x must be multiples of p(x). But then n = p(x)q(x) implies that p(x)isaconstantc.Notethat c=6 1, for that would make I the whole ring, which it is not since 1 ∈= I.Nowwealso have x = cr(x)forsomer(x), which is impossible since c does not divide x (i.e., c has no inverse in Z[x] since the only units are 1). In fact it is easy to see that I is generated by the two elements n and x in the sense of the next theorem. Theorem 6.3. Let R be a commutative ring with identity. Let c1;c2;:::;cn ∈ R.Then the set I = {r1c1 + ···+rncn |r1;:::;rn ∈R} is an ideal of R. Proof. Homework; generalize the proof of Theorem 6.2. We call the ideal I of Theorem 6.3 the ideal generated by c1;:::;cn and denote it by (c1;c2;:::;cn). Comments. 1. If R does not have an identity, there is a complication in the definition since one wants the elements ci ∈ I (see exercise 31, p. 143). 2. If R is not commutative, one needs multiplication on both sides in the definition of I. 3. Ideals with finitely many generators are called finitely generated ideals. One has to work a bit to find ideals which are not finitely generated and we will avoid them in this course. One example is the ideal generated by all the indeterminates in the polynomial ring R[x1;x2;x3;:::] with infinitely many indeterminates. 3 In analogy to congruence in Z and F [x] we now will build a ring R=I for any ideal I in any ring R.Fora; b ∈ R,wesaya is congruent to b modulo I [and write a ≡ b (mod I)] if a − b ∈ I. Note that when I =(n)⊂Zis the principal ideal generated by n,thena−b∈I ⇐⇒ n | ( a − b ), so this is our old notion of congruence. As before, we require congruence to be an equivalence relation if it is going to work for us, so we check this. Theorem 6.4. Let I be an ideal of a ring R. Congruence modulo I is an equivalence relation. Proof. reflexive: a − a =0∈I since I is a subring. symmetric: Assume a ≡ b (mod I). Then a − b ∈ I.SinceIis a subring, its additive inverse, b − a is also in I,andsob≡a(mod I). transitive: Assume a ≡ b (mod I)andb≡c (mod I). Then a − b ∈ I and b − c ∈ I, hence the sum a − c =(a−b)+(b−c)∈I,soa≡c(mod I). We use this to show that arithmetic works \modulo I". Theorem 6.5. Let I be an ideal of a ring R.Ifa≡b(mod I) and c ≡ d (mod I),then (1) a + c ≡ b + d (mod I); (2) ac ≡ bd (mod I). Proof. (1) (a + c) − (b + d)=(a−b)+(c−d). Since a − b ∈ I and c − d ∈ I,sois (a+c)−(b+d), hence a + c ≡ b + d (mod I). (2) ac−bd = ac−bc+bc−bd =(a−b)c+b(c−d)∈Isince I is closed under multiplication on both sides. Therefore ac ≡ bd (mod I). Looking at this proof, we see that it is multiplication that fails if we have only a left or right ideal that is not 2-sided. The equivalence classes for this relation, are commonly called cosets. What do they look like? The congruence class of a modulo I is { b ∈ R | b ≡ a (mod I) } = { b ∈ R | b − a ∈ I } = { b ∈ R | b − a = i for some i ∈ I } = { b ∈ R | b = a + i for some i ∈ I } = { a + i | i ∈ I }: We denote this coset by a + I. As earlier, we have a ≡ b (mod I)iffa+I=b+I.The set of all cosets of I (congruence classes of R modulo I) will be denoted by R=I. 4 Selected problems from pp. 141{145. 13 (generalized). Let I be an ideal in a ring R with 1. I = R iff I contains a unit. Proof. (=⇒)IfI=R,then1∈Iis a unit in I. ( ⇐= ) Let u ∈ I be a unit. Then there exists v ∈ R with vu = 1. For any r ∈ R,weget r=r·1=r(vu)=(rv)u ∈ I. 14/35. A commutative ring R with 1 is a field iff its only two ideals are (0) and R. Proof. (=⇒) Any nonzero ideal I contains some nonzero element, which is a unit since R is a field. By #13, I = R. ( ⇐= ) Let 0 =6 a ∈ R and let I =(a). By hypothesis, I = R,soIcontains the identity 1. Therefore 1 = ra for some r ∈ R,sothatris the inverse of a. Therefore R is a field. 38. Every ideal I in Z is principal. Proof. Assume I =6 (0) (which is principal). Let c be the smallest positive element in I (exists by the well-ordering axiom). Then (c) ⊆ I. Conversely, let a ∈ I. By the division algorithm, we can write a = cq + r with 0 ≤ r<c. Then r = a − cq ∈ I. By our choice of c,wemusthaver= 0, as otherwise it is a smaller positive element of I. Therefore a ∈ (c), so I =(c) is principal. m 39. (a) S = { n | m; n ∈ Z;n odd } is a subring of Q. { m ∈ | } (b) I = n S m even is an ideal in S. (c) S=I has exactly two cosets. Proof. (a) Check closure under subtraction and multiplication. (b) Check closure under subtraction and multiplication by elements of S. m m m−n (c) If n ∈= I, then, since m and n are both odd, we see that n =1+ n ∈1+I.Sothe only cosets are I and 1 + I. 5 Quotient rings and homomorphisms. Theorem 6.5 gives the fact that addition and multiplication are well-defined on congru- ence classes. Translating this into the language of cosets gives Theorem 6.8.
Load more

Recommended publications
  • Scott Spaces and the Dcpo Category
    SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology.
    [Show full text]
  • A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints
    A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints by Mehrdad Sabetzadeh A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Computer Science University of Toronto Copyright c 2003 by Mehrdad Sabetzadeh Abstract A Category-Theoretic Approach to Representation and Analysis of Inconsistency in Graph-Based Viewpoints Mehrdad Sabetzadeh Master of Science Graduate Department of Computer Science University of Toronto 2003 Eliciting the requirements for a proposed system typically involves different stakeholders with different expertise, responsibilities, and perspectives. This may result in inconsis- tencies between the descriptions provided by stakeholders. Viewpoints-based approaches have been proposed as a way to manage incomplete and inconsistent models gathered from multiple sources. In this thesis, we propose a category-theoretic framework for the analysis of fuzzy viewpoints. Informally, a fuzzy viewpoint is a graph in which the elements of a lattice are used to specify the amount of knowledge available about the details of nodes and edges. By defining an appropriate notion of morphism between fuzzy viewpoints, we construct categories of fuzzy viewpoints and prove that these categories are (finitely) cocomplete. We then show how colimits can be employed to merge the viewpoints and detect the inconsistencies that arise independent of any particular choice of viewpoint semantics. Taking advantage of the same category-theoretic techniques used in defining fuzzy viewpoints, we will also introduce a more general graph-based formalism that may find applications in other contexts. ii To my mother and father with love and gratitude. Acknowledgements First of all, I wish to thank my supervisor Steve Easterbrook for his guidance, support, and patience.
    [Show full text]
  • The Structure Theory of Complete Local Rings
    The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings.
    [Show full text]
  • Formal Power Series - Wikipedia, the Free Encyclopedia
    Formal power series - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Formal_power_series Formal power series From Wikipedia, the free encyclopedia In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates. This perspective contrasts with that of power series, whose variables designate numerical values, and which series therefore only have a definite value if convergence can be established. Formal power series are often used merely to represent the whole collection of their coefficients. In combinatorics, they provide representations of numerical sequences and of multisets, and for instance allow giving concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved; this is known as the method of generating functions. Contents 1 Introduction 2 The ring of formal power series 2.1 Definition of the formal power series ring 2.1.1 Ring structure 2.1.2 Topological structure 2.1.3 Alternative topologies 2.2 Universal property 3 Operations on formal power series 3.1 Multiplying series 3.2 Power series raised to powers 3.3 Inverting series 3.4 Dividing series 3.5 Extracting coefficients 3.6 Composition of series 3.6.1 Example 3.7 Composition inverse 3.8 Formal differentiation of series 4 Properties 4.1 Algebraic properties of the formal power series ring 4.2 Topological properties of the formal power series
    [Show full text]
  • Math 412. Quotient Rings of Polynomial Rings Over a Field
    (c)Karen E. Smith 2018 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 412. Quotient Rings of Polynomial Rings over a Field. For this worksheet, F always denotes a fixed field, such as Q; R; C; or Zp for p prime. DEFINITION. Fix a polynomial f(x) 2 F[x]. Define two polynomials g; h 2 F[x] to be congruent modulo f if fj(g − h). We write [g]f for the set fg + kf j k 2 F[x]g of all polynomials congruent to g modulo f. We write F[x]=(f) for the set of all congruence classes of F[x] modulo f. The set F[x]=(f) has a natural ring structure called the Quotient Ring of F[x] by the ideal (f). CAUTION: The elements of F[x]=(f) are sets so the addition and multiplication, while induced by those in F[x], is a bit subtle. A. WARM UP. Fix a polynomial f(x) 2 F[x] of degree d > 0. (1) Discuss/review what it means that congruence modulo f is an equivalence relation on F[x]. (2) Discuss/review with your group why every congruence class [g]f contains a unique polynomial of degree less than d. What is the analogous fact for congruence classes in Z modulo n? 2 (3) Show that there are exactly four distinct congruence classes for Z2[x] modulo x + x. List them out, in set-builder notation. For each of the four, write it in the form [g]x2+x for two different g.
    [Show full text]
  • Quotient-Rings.Pdf
    4-21-2018 Quotient Rings Let R be a ring, and let I be a (two-sided) ideal. Considering just the operation of addition, R is a group and I is a subgroup. In fact, since R is an abelian group under addition, I is a normal subgroup, and R the quotient group is defined. Addition of cosets is defined by adding coset representatives: I (a + I)+(b + I)=(a + b)+ I. The zero coset is 0+ I = I, and the additive inverse of a coset is given by −(a + I)=(−a)+ I. R However, R also comes with a multiplication, and it’s natural to ask whether you can turn into a I ring by multiplying coset representatives: (a + I) · (b + I)= ab + I. I need to check that that this operation is well-defined, and that the ring axioms are satisfied. In fact, everything works, and you’ll see in the proof that it depends on the fact that I is an ideal. Specifically, it depends on the fact that I is closed under multiplication by elements of R. R By the way, I’ll sometimes write “ ” and sometimes “R/I”; they mean the same thing. I Theorem. If I is a two-sided ideal in a ring R, then R/I has the structure of a ring under coset addition and multiplication. Proof. Suppose that I is a two-sided ideal in R. Let r, s ∈ I. Coset addition is well-defined, because R is an abelian group and I a normal subgroup under addition. I proved that coset addition was well-defined when I constructed quotient groups.
    [Show full text]
  • Modular Arithmetic
    CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 5 Modular Arithmetic One way to think of modular arithmetic is that it limits numbers to a predefined range f0;1;:::;N ¡ 1g, and wraps around whenever you try to leave this range — like the hand of a clock (where N = 12) or the days of the week (where N = 7). Example: Calculating the day of the week. Suppose that you have mapped the sequence of days of the week (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday) to the sequence of numbers (0;1;2;3;4;5;6) so that Sunday is 0, Monday is 1, etc. Suppose that today is Thursday (=4), and you want to calculate what day of the week will be 10 days from now. Intuitively, the answer is the remainder of 4 + 10 = 14 when divided by 7, that is, 0 —Sunday. In fact, it makes little sense to add a number like 10 in this context, you should probably find its remainder modulo 7, namely 3, and then add this to 4, to find 7, which is 0. What if we want to continue this in 10 day jumps? After 5 such jumps, we would have day 4 + 3 ¢ 5 = 19; which gives 5 modulo 7 (Friday). This example shows that in certain circumstances it makes sense to do arithmetic within the confines of a particular number (7 in this example), that is, to do arithmetic by always finding the remainder of each number modulo 7, say, and repeating this for the results, and so on.
    [Show full text]
  • Kernel Methods for Knowledge Structures
    Kernel Methods for Knowledge Structures Zur Erlangung des akademischen Grades eines Doktors der Wirtschaftswissenschaften (Dr. rer. pol.) von der Fakultät für Wirtschaftswissenschaften der Universität Karlsruhe (TH) genehmigte DISSERTATION von Dipl.-Inform.-Wirt. Stephan Bloehdorn Tag der mündlichen Prüfung: 10. Juli 2008 Referent: Prof. Dr. Rudi Studer Koreferent: Prof. Dr. Dr. Lars Schmidt-Thieme 2008 Karlsruhe Abstract Kernel methods constitute a new and popular field of research in the area of machine learning. Kernel-based machine learning algorithms abandon the explicit represen- tation of data items in the vector space in which the sought-after patterns are to be detected. Instead, they implicitly mimic the geometry of the feature space by means of the kernel function, a similarity function which maintains a geometric interpreta- tion as the inner product of two vectors. Knowledge structures and ontologies allow to formally model domain knowledge which can constitute valuable complementary information for pattern discovery. For kernel-based machine learning algorithms, a good way to make such prior knowledge about the problem domain available to a machine learning technique is to incorporate it into the kernel function. This thesis studies the design of such kernel functions. First, this thesis provides a theoretical analysis of popular similarity functions for entities in taxonomic knowledge structures in terms of their suitability as kernel func- tions. It shows that, in a general setting, many taxonomic similarity functions can not be guaranteed to yield valid kernel functions and discusses the alternatives. Secondly, the thesis addresses the design of expressive kernel functions for text mining applications. A first group of kernel functions, Semantic Smoothing Kernels (SSKs) retain the Vector Space Models (VSMs) representation of textual data as vec- tors of term weights but employ linguistic background knowledge resources to bias the original inner product in such a way that cross-term similarities adequately con- tribute to the kernel result.
    [Show full text]
  • Right Ideals of a Ring and Sublanguages of Science
    RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance.
    [Show full text]
  • 6. Localization
    52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i.
    [Show full text]
  • 2.2 Kernel and Range of a Linear Transformation
    2.2 Kernel and Range of a Linear Transformation Performance Criteria: 2. (c) Determine whether a given vector is in the kernel or range of a linear trans- formation. Describe the kernel and range of a linear transformation. (d) Determine whether a transformation is one-to-one; determine whether a transformation is onto. When working with transformations T : Rm → Rn in Math 341, you found that any linear transformation can be represented by multiplication by a matrix. At some point after that you were introduced to the concepts of the null space and column space of a matrix. In this section we present the analogous ideas for general vector spaces. Definition 2.4: Let V and W be vector spaces, and let T : V → W be a transformation. We will call V the domain of T , and W is the codomain of T . Definition 2.5: Let V and W be vector spaces, and let T : V → W be a linear transformation. • The set of all vectors v ∈ V for which T v = 0 is a subspace of V . It is called the kernel of T , And we will denote it by ker(T ). • The set of all vectors w ∈ W such that w = T v for some v ∈ V is called the range of T . It is a subspace of W , and is denoted ran(T ). It is worth making a few comments about the above: • The kernel and range “belong to” the transformation, not the vector spaces V and W . If we had another linear transformation S : V → W , it would most likely have a different kernel and range.
    [Show full text]
  • 23. Kernel, Rank, Range
    23. Kernel, Rank, Range We now study linear transformations in more detail. First, we establish some important vocabulary. The range of a linear transformation f : V ! W is the set of vectors the linear transformation maps to. This set is also often called the image of f, written ran(f) = Im(f) = L(V ) = fL(v)jv 2 V g ⊂ W: The domain of a linear transformation is often called the pre-image of f. We can also talk about the pre-image of any subset of vectors U 2 W : L−1(U) = fv 2 V jL(v) 2 Ug ⊂ V: A linear transformation f is one-to-one if for any x 6= y 2 V , f(x) 6= f(y). In other words, different vector in V always map to different vectors in W . One-to-one transformations are also known as injective transformations. Notice that injectivity is a condition on the pre-image of f. A linear transformation f is onto if for every w 2 W , there exists an x 2 V such that f(x) = w. In other words, every vector in W is the image of some vector in V . An onto transformation is also known as an surjective transformation. Notice that surjectivity is a condition on the image of f. 1 Suppose L : V ! W is not injective. Then we can find v1 6= v2 such that Lv1 = Lv2. Then v1 − v2 6= 0, but L(v1 − v2) = 0: Definition Let L : V ! W be a linear transformation. The set of all vectors v such that Lv = 0W is called the kernel of L: ker L = fv 2 V jLv = 0g: 1 The notions of one-to-one and onto can be generalized to arbitrary functions on sets.
    [Show full text]