0.2 Vector spaces. J.A.Beachy 1


 Kathlyn Ferguson
 3 years ago
 Views:
Transcription
1 J.A.Beachy Vector spaces I m going to begin this section at a rather basic level, giving the definitions of a field and of a vector space in much that same detail as you would have met them in a first linear algebra course. If you took the course at the sophomore level, you may have only used scalars from the field R of real numbers. We will allow the scalars to come from any field, but it isn t too much of a jump. You should remember, of course, that a field can be finite (keep Z 2 in mind). Most of the statements are taken from the appendices in the text, and are numbered accordingly. DEFINITION A.1.1. A field is a set F together with two operations + and for which the following conditions hold: (i) (Closure ) for all a, b F, the sum a + b and the product a b again belong to F ; (ii) (Associativity ) for all a, b, c F, a + (b + c) = (a + b) + c and a (b c) = (a b) c; (iii) (Commutativity ) For all a, b F, a + b = b + a and a b = b a; (iv) (Distributive laws ) for all a, b, c F, a (b+c) = a b+a c and (a+b) c = a c+b c; (v) (Existence of an additive identity ) there exists an element 0 F for which a+0 = a and 0 + a = a, for all a F ; (vi) (Existence of a multiplicative identity ) there exists an element 1 F, with 1 0, for which a 1 = a and 1 a = a, for all a F ; (vii) (Existence of additive inverses ) for each a F, the equations a + x = 0 and x + a = 0 have a solution x in F, called the additive inverse of a, and denoted by a; (viii) (Existence of multiplicative inverses ) for each a F, with a 0, the equations a x = 1 and x a = 1 have a solution x in F, called the multiplicative inverse of a, and denoted by a 1. In any field, we can identify the integer n with the element n 1. Although we can divide by any nonzero element of the field, we cannot necessarily divide by a nonzero integer since it is possible that n 1 = 0 even though n 0. To deal with this problem we consider the characteristic of the field. (For details see Definition A.1.2 and Proposition A.1.3 in the text, together with Examples and ) Our goal is to study rings, and so we want to include the ring Z of integers, as well as the ring F [x] = {a 0 + a 1 x + a 2 x a n 1 x n 1 + a n x n a i F, 0 i n} of all polynomials in one variable with coefficients in a field F. In Z, only 1 and 1 have multiplicative inverses; in F [x], only the nonzero constant polynomials have multiplicative inverses. In the definition of a ring we will keep all but condition (viii), which requires that each nonzero element must have a multiplicative inverse. Dropping this one condition gives the definition of a commutative ring with identity, stated in Definition (a). Even within linear algebra it is crucial to study matrices. In our study of rings, we want to include rings whose elements are matrices, and so we have to give up half of condition (iii) in the definition of a field, which is the requirement that multiplication must satisfy
2 2 J.A.Beachy RINGS AND MODULES: CLASS NOTES the commutative law. Dropping both commutativity of multiplication and the existence of multiplicative inverses gives Definition 1.1.1, which defines an associative ring with identity. In the definition of a field, if we drop only the commutativity of multiplication, we arrive at the notion of a skew field, also called a division ring (see Definition (b)). Modules over skew fields turn out to have most of the properties of vector spaces over a field. Here are some additional properties that follow immediately from the definition of a field. Actually, they hold in any ring, but we simply state them here for any elements a, b, c in the field F. (a) If a + c = b + c, then a = b. (b) If a c = b c and c 0, then a = b. (c) a 0 = 0 (d) ( a) = a (e) ( a) ( b) = a b DEFINITION A.1.4. If F and E are fields such that F is a subset of E and the operations on F are those induced by E, then we say that F is a subfield of E and that E is an extension field of F. The next definition we need to review is that of a vector space over the field F. Our ultimate goal is to study the analog over a ring R, which we will call a left Rmodule. If you make a careful comparison between Definitions and A.1.5, you will see that we can use exactly the same axioms in both cases. The only change is that the scalars are allowed to come from a ring instead of a field. Of coruse, this does make things much more complicated, because we lose the ability to divide through be a scalar. If you don t want to wait for examples of modules, you can skip ahead in the text to examples and Example take note of the fact that the ring R is a module over itself; Example just shows that a vector space is an example of a module; Example shows how to think of an abelian group A as a module over the ring Z by defining a scalar multiplication n a in terms of the addition in A. You may also find it interesting to read Example 2.1.5, which shows that the vector space over F defined by using all column vectors with n components is actually a module over the ring M n (F ) of all n n matrices over F. To see why there is a scalar multiplication, you can think of the column vector as a matrix, and then you can multiply it on the left by an n n matrix. DEFINITION A.1.5. A vector space over the field F is a set V on which two operations are defined, called addition and scalar multiplication, and denoted by + and respectively. The operations must satisfy the following conditions: (i) (Closure ) for all a F and all u, v V, the sum u + v and the scalar product a v are uniquely defined and belong to V ; (ii) (Associativity ) for all a, b F and all u, v, w V, u + (v + w) = (u + v) + w and a (b v) = (a b) v; (iii) (Commutativity of addition ) for all u, v V, u + v = v + u; (iv) (Distributive laws ) for all a, b F and all u, v V, a (u + v) = (a u) + (a v) and (a + b) v = (a v) + (b v); (v) (Existence of an additive identity ) there exists an element 0 in V for which v+0 = v and 0 + v = v for all v V ;
3 0.2. VECTOR SPACES J.A.Beachy 3 (vi) (Existence of additive inverses ) for each v V, the equations v+x = 0 and x+v = 0 have a solution x V, denoted by v; (vii) (Unitary law ) for all v V, 1 v = v. Using 0 for the additive identity of V, and v for the additive inverse of v V, we have the following results: 0 + v = v, a 0 = 0, and ( a)v = a( v) = (av) for all a F and v V. The proofs involve the distributive laws, which give the only connection between addition and scalar multiplication. One of the most basic examples of a vector space (which turns out to be very important in Galois theory) comes up in the situation in which E is an extension field of the field F. Then E is certainly an abelian group, and because F is a subfield of E it is possible to multiply elements of E by elements of F. This shows that E is actually a vector space over F. The next example is probably the one that is the most familiar to you. EXAMPLE. For any field F, the set F n of ntuples is a vector space over F. The ntuples can be written either as a row vector or as a column vector. Addition of ntuples is defined component by component: (x 1, x 2,..., x n ) + (y 1, y 2,..., y n ) = (x 1 + y 1, x 2 + y 2,..., x n + y n ). Similarly, scalar multiplication is defined componentwise: a (x 1, x 2,..., x n ) = (ax 1, ax 2,..., ax n ). It is not hard to check that all of the necessary axioms are satisfied. In each case it comes down to the fact that the field axioms hold in each component. The construction in the example, using ntuples, can also be given for modules. In this case it is called a direct sum of modules. (See Section 2.2 of the text for the definition and a discussion of direct sums of modules.) DEFINITION. Given a vector space V over a field F, a subset W of V is called a subspace if W is a vector space over F under the operations already defined on V. PROPOSITION. A subset W of a vector space V is a subspace of V iff (i) W is nonempty; (ii) if u, v W, then u + v W ; and (iii) if v W and a F, then av W. There is a corresponding notion of a submodule of a module, with the obvious definition. In the case of a ring R, which we can view as a module over itself, any submodule is called a left ideal. The ring R is also a right module over itself, because we can multiply by the scalars on the right as well as the left. In this case we get the notion of a right ideal, and then a subset of R that is both a left ideal and a right ideal is called a twosided ideal, or simply an ideal. These definitions are given on page 68 of the text. After defining the notions of vector spaces and subspaces, the next step is to identify the functions that can be used to relate one vector space to another. These functions should
4 4 J.A.Beachy RINGS AND MODULES: CLASS NOTES respect the algebraic structure of the vector spaces, so it is reasonable to require that they preserve addition and scalar multiplication. DEFINITION A Let V and W be vector spaces over the field F. A linear transformation from V to W is a function f : V W such that f(au + bv) = af(u) + bf(v) for all vectors u, v V and all scalars a, b F. If a linear transformation is onetoone and onto, it is called a vector space isomorphism, or simply an isomorphism. You can think of a vector space isomorphism f : V W as just giving a way to rename the elements of V so that they look just like the elements of W, since any isomorphism preserves all of the algebraic structure. One of the powerful ideas of abstract algebra is to think of isomorphic objects as simply being one and the same. Corresponding to the definition of a linear transformations between vector spaces, we have the concept of an Rhomomorphism between left Rmodules. You can see, in Definition 2.1.6, that to state the general definition we only have to change the field F to a ring R. Of course, this rather innocuous change causes a great many problems. Or maybe I should just say that it makes life much more interesting. With this additional definition, you can read the next example in the text, Example 2.1.6, which turns out to have very important implications for linear algebra. This example starts with a vector space V over a field F, and a single linear transformation T : V V. To help understand what T is doing, we can think of the action of T as a multiplication, so that we define T v = T (v), for every v V. But from our point of view, a multiplication like this should be defined for an entire ring, not just for a single element T. We can form powers T n, using composition of functions, and linear combinations a 0 I + a 1 T + + a n T n, where a i F and I is the identity transformation on V. These can be made into a ring, and used for the scalars, but it is easier to let the ring of scalars be the polynomial ring F [x]; we just have to substitute T in place of x. These remarks lead to the multiplication in Example 2.1.6: given a polynomial a 0 + a 1 x + + a n x n F [x] and a vector v V, we define (a 0 + a 1 x + + a n x n ) v = [a 0 I + a 1 T + + a n T n ](v) = a 0 v + a 1 T (v) + + a n T n (v). This multiplication, together with a structure theorem for certain modules, can be used to find standard forms for the matrix associated with T (see Section 2.7). We next turn to the definition of a basis. The fact every vector space has a basis is certainly the single most important property of vector spaces. DEFINITION A.1.6 (a). Let S = {v 1,..., v n } be a set of vectors in the vector space V over the field F. Any vector of the form v = n i=1 a iv i, for a i F, is called a linear combination of the vectors in S. The set S is said to span V if each element of V can be expressed as a linear combination of the vectors in S. DEFINITION A.1.6 (b). Let S = {v 1,..., v n } be a set of vectors in the vector space V over the field F. The vectors in S are said to be linearly dependent if one of the vectors can
5 0.2. VECTOR SPACES J.A.Beachy 5 be expressed as a linear combination of the others. If not, then S is said to be a linearly independent set. DEFINITION A.1.9 (a). A subset of the vector space V is called a basis for V if it spans V and is linearly independent. PROPOSITION A Let S be a nonempty subset of the vector space V. Then S is a basis for V if and only if each vector in V can be written uniquely as a linear combination of vectors in S. THEOREM. Every vector space has a basis. This important theorem is proved in the text in Theorem A.2.5, as an illustration of the use of Zorn s lemma. As a consequence of more general results for modules, it is proved again in Corollary of the text, which contains the stronger result that any nonzero vector space over a skew field has a basis. The notion of a basis can be generalized to Rmodules, as in Definition 2.2.1, and a module is called a free module if it has a basis. One of the important differences between modules and vector spaces is that a module need not be free. It is not hard to find examples of modules that are not free, since most modules tend to be a long way from having a basis. Here is the most elementary example of a module that is not free. Let R be the ring Z 4 = {0, 1, 2, 3}. The subset M = {0, 2} is closed under addition and multiplication by any element of R, so it is a submodule of R, when R is thought of as a module over itself. No basis can contain 0, so the only possibility for a basis for M is the set {2}. But if M had a basis consisting of a single element, then M would be forced to have the same number of elements as R, and this clearly isn t the case. You can see that M is a perfectly good module, but it is too small to be a free module over this particular ring. Actually, if all nonzero left Rmodules are free it forces R to be a skew field (see the exercises in Section 2.3 of the notes). Although it is rare for a module to have a basis, it always has a spanning set. Even though in most cases we can t find an independent spanning set, it is still important to know when we can find a finite spanning set. In Definition (b) a module is said to be finitely generated if it has a finite spanning set. If it can be spanned by a single element, it is called cyclic (see Definition (a)). One of the famous theorems that is proved in the text is the Hilbert basis theorem (Theorem ), which in its more elementary form states that if F is a field, then over the polynomial ring F [x 1, x 2,..., x n ], every submodule of a finitely generated module is again finitely generated. The next theorem makes an even stronger statement than just asserting the existence of a basis in any vector space. It shows that any linearly independent set of vectors can be extended to a basis for the vector space. As an important consequence, we get Theorem A.2.6, which shows that every subspace of a vector space has a complement. THEOREM A.2.5. Let F be a field, and let V be any vector space over F. Then every linearly independent subset of V is contained in a basis for V.
6 6 J.A.Beachy RINGS AND MODULES: CLASS NOTES THEOREM A.2.6. Let V be a vector space over the field F, and let W be any subspace of V. Then there exists a subspace Y of V such that each element v V can be written uniquely in the form v = w + y for some w W and y Y. Once again there are problems when we move from vector spaces to modules, because we can lose the existence of complements. For example, the submodule (or ideal) M = {0, 2} in the ring Z 4 does not have a complement. To see this, just note that there is no other proper nonzero submodule, since 1 and 3 are invertible in the ring, and therefore can t belong to a proper submodule. Thus Theorem A.2.6 raises an important question for modules. Can we find conditions on the ring that guarantee that every proper submodule of every nonzero module has a complement? Modules that have this crucial property are called completely reducible in Definition The answer to the question is contained in the ArtinWedderburn theorem (see Theorem 3.3.2, Corollary 3.3.4, and Theorem 2.3.6), which shows that every nonzero left Rmodule is completely reducible if and only if the ring R can be written as a finite direct sum of matrix rings over skew fields. Our next step is to look at the concept of the dimension of a vector space. Once again, it turns out that life is much more complicated for modules over a ring than it is for vector spaces over a field. But modules do have various weaker properties that can still be used to prove some of the results that hold for vector spaces. THEOREM A.1.7. Let V be a vector space, let S = {u 1, u 2,..., u m } be a set that spans V, and let T = {v 1, v 2,..., v n } be a linearly independent set. Then n m, and V can be spanned by a set of m vectors that contains T. COROLLARY A.1.8. Any two finite subsets that both span V and are linearly independent must have the same number of elements. DEFINITION A.1.9 (b). If V has a finite basis, then it is said to be finite dimensional, and the number of vectors in the basis is called the dimension of V, denoted by dim(v ). THEOREM. Any ndimensional vector space over the field F is isomorphic to F n. EXAMPLE. The field C of complex numbers is a two dimensional vector space over the field R. EXAMPLE. If F is a field, then the set of polynomials F [x] with coefficients in F is an infinite dimensional vector space over F. The above theorem stating that any ndimensional vector space is isomorphic to F n has a direct generalization to modules. It is proved in Proposition (a) that any Rmodule with a basis consisting of n elements is isomorphic to the module of ntuples of elements from R, denoted by R n, where addition and scalar multiplication are defined component by component.
7 0.2. VECTOR SPACES J.A.Beachy 7 It is Corollary A.1.8 that justifies the definition of the dimension of a vector space over a field, since it shows that the number of elements in a basis is an invariant of the vector space. When we look at free modules over a ring, this important result may fail to hold. The ring R is said to have invariant basis number if the following condition holds: if M is a left R module with a basis consisting of n elements, then every other basis for M also has n elements. There are exercises in the notes that look at this question see exercise sets in Section 2.5 and Section 2.6. In particular, the second of these exercises gives an example of a ring that does not have invariant basis number. Now I want to discuss some of the good properties of vector spaces that come from the ability to give a definition of dimension. First of all, if a vector space V has finite dimension, then so does each of its nonzero subspaces. The next proposition follows from Theorem A.2.5. PROPOSITION. If V is a vector space with dim(v ) = n, then dim(w ) < n for any proper subspace W. As a consequence of the proposition, if V 0 V 1 V 2 V k 1 V k is any chain of distinct subspaces of V, and dim(v ) = n <, then we must have k n. For example, if dim(v ) = 1, then the only subspaces are V and {0}, and V {0} is the longest chain of distinct subspaces. The onedimensional subspaces play an important role for vector spaces, since any onedimensional vector space is just a copy of the field. To translate to modules, we focus on the fact that a onedimensional vector space has no proper nonzero submodules. The general definition is given in Definition 2.1.9: a nonzero module M is called simple if its only submodules are M and {0}. To see how simple modules can be described explicitly in terms of the ring, refer to Proposition (a) and Proposition (a). The simple modules play an important role, especially in representation theory. Over a field, every vector space is isomorphic to a direct sum of onedimensional subspaces. Over a ring, it need not be true that every module is isomorphic to a direct sum of simple submodules. (Just look at Z 4 as a module over itself.) According to Definition (b), a module is called semisimple if it is isomorphic to a direct sum of simple submodules. As something of a surprise, it turns out that semisimple modules are the same as completely reducible modules, as shown by Corollary This section has previewed a long list of definitions, much too long to remember the first time you read it. I hope that you will come back to this section as you meet these definitions in the text, so that you can keep the new theory in perspective. It will also help to keep in mind what happens for abelian groups, and that is the next topic.
Part IV. Rings and Fields
IV.18 Rings and Fields 1 Part IV. Rings and Fields Section IV.18. Rings and Fields Note. Roughly put, modern algebra deals with three types of structures: groups, rings, and fields. In this section we
More informationa (b + c) = a b + a c
Chapter 1 Vector spaces In the Linear Algebra I module, we encountered two kinds of vector space, namely real and complex. The real numbers and the complex numbers are both examples of an algebraic structure
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationRings. EE 387, Notes 7, Handout #10
Rings EE 387, Notes 7, Handout #10 Definition: A ring is a set R with binary operations, + and, that satisfy the following axioms: 1. (R, +) is a commutative group (five axioms) 2. Associative law for
More information(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d
The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers
More information1 Last time: inverses
MATH Linear algebra (Fall 8) Lecture 8 Last time: inverses The following all mean the same thing for a function f : X Y : f is invertible f is onetoone and onto 3 For each b Y there is exactly one a
More information1 Fields and vector spaces
1 Fields and vector spaces In this section we revise some algebraic preliminaries and establish notation. 1.1 Division rings and fields A division ring, or skew field, is a structure F with two binary
More information2 so Q[ 2] is closed under both additive and multiplicative inverses. a 2 2b 2 + b
. FINITEDIMENSIONAL VECTOR SPACES.. Fields By now you ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices,
More information3.1 Definition of a Group
3.1 J.A.Beachy 1 3.1 Definition of a Group from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair This section contains the definitions of a binary operation,
More informationGRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.
GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196  INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups  Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationVector spaces. EE 387, Notes 8, Handout #12
Vector spaces EE 387, Notes 8, Handout #12 A vector space V of vectors over a field F of scalars is a set with a binary operator + on V and a scalarvector product satisfying these axioms: 1. (V, +) is
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationChapter 2: Linear Independence and Bases
MATH20300: Linear Algebra 2 (2016 Chapter 2: Linear Independence and Bases 1 Linear Combinations and Spans Example 11 Consider the vector v (1, 1 R 2 What is the smallest subspace of (the real vector space
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the socalled prime and maximal ideals. Let
More informationALGEBRA EXERCISES, PhD EXAMINATION LEVEL
ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)
More informationReview of Linear Algebra
Review of Linear Algebra Throughout these notes, F denotes a field (often called the scalars in this context). 1 Definition of a vector space Definition 1.1. A F vector space or simply a vector space
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked
More informationSpanning, linear dependence, dimension
Spanning, linear dependence, dimension In the crudest possible measure of these things, the real line R and the plane R have the same size (and so does 3space, R 3 ) That is, there is a function between
More informationRings If R is a commutative ring, a zero divisor is a nonzero element x such that xy = 0 for some nonzero element y R.
Rings 10262008 A ring is an abelian group R with binary operation + ( addition ), together with a second binary operation ( multiplication ). Multiplication must be associative, and must distribute over
More informationAbstract & Applied Linear Algebra (Chapters 12) James A. Bernhard University of Puget Sound
Abstract & Applied Linear Algebra (Chapters 12) James A. Bernhard University of Puget Sound Copyright 2018 by James A. Bernhard Contents 1 Vector spaces 3 1.1 Definitions and basic properties.................
More informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More informationDefinition 1. A set V is a vector space over the scalar field F {R, C} iff. there are two operations defined on V, called vector addition
6 Vector Spaces with Inned Product Basis and Dimension Section Objective(s): Vector Spaces and Subspaces Linear (In)dependence Basis and Dimension Inner Product 6 Vector Spaces and Subspaces Definition
More informationFirst we introduce the sets that are going to serve as the generalizations of the scalars.
Contents 1 Fields...................................... 2 2 Vector spaces.................................. 4 3 Matrices..................................... 7 4 Linear systems and matrices..........................
More informationMath 110, Spring 2015: Midterm Solutions
Math 11, Spring 215: Midterm Solutions These are not intended as model answers ; in many cases far more explanation is provided than would be necessary to receive full credit. The goal here is to make
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationAbstract Vector Spaces
CHAPTER 1 Abstract Vector Spaces 1.1 Vector Spaces Let K be a field, i.e. a number system where you can add, subtract, multiply and divide. In this course we will take K to be R, C or Q. Definition 1.1.
More informationAugust 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei. 1.
August 23, 2017 Let us measure everything that is measurable, and make measurable everything that is not yet so. Galileo Galilei 1. Vector spaces 1.1. Notations. x S denotes the fact that the element x
More informationSolution to Homework 1
Solution to Homework Sec 2 (a) Yes It is condition (VS 3) (b) No If x, y are both zero vectors Then by condition (VS 3) x = x + y = y (c) No Let e be the zero vector We have e = 2e (d) No It will be false
More informationDefinition 2.3. We define addition and multiplication of matrices as follows.
14 Chapter 2 Matrices In this chapter, we review matrix algebra from Linear Algebra I, consider row and column operations on matrices, and define the rank of a matrix. Along the way prove that the row
More informationSubfields of division algebras. J. Bell
Subfields of division algebras J. Bell 1 In this talk, we consider domains which are finitely generated over an algebraically closed field F. Given such a domain A, we know that if A does not contain a
More information* 8 Groups, with Appendix containing Rings and Fields.
* 8 Groups, with Appendix containing Rings and Fields Binary Operations Definition We say that is a binary operation on a set S if, and only if, a, b, a b S Implicit in this definition is the idea that
More informationA Little Beyond: Linear Algebra
A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of nonzero polynomials in [x], no two
More informationInfiniteDimensional Triangularization
InfiniteDimensional Triangularization Zachary Mesyan March 11, 2018 Abstract The goal of this paper is to generalize the theory of triangularizing matrices to linear transformations of an arbitrary vector
More informationMAT2342 : Introduction to Applied Linear Algebra Mike Newman, fall Projections. introduction
MAT4 : Introduction to Applied Linear Algebra Mike Newman fall 7 9. Projections introduction One reason to consider projections is to understand approximate solutions to linear systems. A common example
More informationChapter 1 Vector Spaces
Chapter 1 Vector Spaces PerOlof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 110 Linear Algebra Vector Spaces Definition A vector space V over a field
More informationLinear algebra and differential equations (Math 54): Lecture 10
Linear algebra and differential equations (Math 54): Lecture 10 Vivek Shende February 24, 2016 Hello and welcome to class! As you may have observed, your usual professor isn t here today. He ll be back
More informationALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS.
ALGEBRA II: RINGS AND MODULES OVER LITTLE RINGS. KEVIN MCGERTY. 1. RINGS The central characters of this course are algebraic objects known as rings. A ring is any mathematical structure where you can add
More informationTuneUp Lecture Notes Linear Algebra I
TuneUp Lecture Notes Linear Algebra I One usually first encounters a vector depicted as a directed line segment in Euclidean space, or what amounts to the same thing, as an ordered ntuple of numbers
More informationis an isomorphism, and V = U W. Proof. Let u 1,..., u m be a basis of U, and add linearly independent
Lecture 4. GModules PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 4. The categories of Gmodules, mostly for finite groups, and a recipe for finding every irreducible Gmodule of a
More informationNOTES (1) FOR MATH 375, FALL 2012
NOTES 1) FOR MATH 375, FALL 2012 1 Vector Spaces 11 Axioms Linear algebra grows out of the problem of solving simultaneous systems of linear equations such as 3x + 2y = 5, 111) x 3y = 9, or 2x + 3y z =
More informationAbstract Vector Spaces and Concrete Examples
LECTURE 18 Abstract Vector Spaces and Concrete Examples Our discussion of linear algebra so far has been devoted to discussing the relations between systems of linear equations, matrices, and vectors.
More informationChapter 4 Vector Spaces And Modules
Chapter 4 Vector Spaces And Modules Up to this point we have been introduced to groups and to rings; the former has its motivation in the set of onetoone mappings of a set onto itself, the latter, in
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationchapter 12 MORE MATRIX ALGEBRA 12.1 Systems of Linear Equations GOALS
chapter MORE MATRIX ALGEBRA GOALS In Chapter we studied matrix operations and the algebra of sets and logic. We also made note of the strong resemblance of matrix algebra to elementary algebra. The reader
More informationDSGA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DSGA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More informationNote that a unit is unique: 1 = 11 = 1. Examples: Nonnegative integers under addition; all integers under multiplication.
Algebra fact sheet An algebraic structure (such as group, ring, field, etc.) is a set with some operations and distinguished elements (such as 0, 1) satisfying some axioms. This is a fact sheet with definitions
More informationLinear Algebra. Chapter 5
Chapter 5 Linear Algebra The guiding theme in linear algebra is the interplay between algebraic manipulations and geometric interpretations. This dual representation is what makes linear algebra a fruitful
More informationMath 54 HW 4 solutions
Math 54 HW 4 solutions 2.2. Section 2.2 (a) False: Recall that performing a series of elementary row operations A is equivalent to multiplying A by a series of elementary matrices. Suppose that E,...,
More informationMA106 Linear Algebra lecture notes
MA106 Linear Algebra lecture notes Lecturers: Diane Maclagan and Damiano Testa 201718 Term 2 Contents 1 Introduction 3 2 Matrix review 3 3 Gaussian Elimination 5 3.1 Linear equations and matrices.......................
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationMath 346 Notes on Linear Algebra
Math 346 Notes on Linear Algebra Ethan Akin Mathematics Department Fall, 2014 1 Vector Spaces Anton Chapter 4, Section 4.1 You should recall the definition of a vector as an object with magnitude and direction
More informationAnswers in blue. If you have questions or spot an error, let me know. 1. Find all matrices that commute with A =. 4 3
Answers in blue. If you have questions or spot an error, let me know. 3 4. Find all matrices that commute with A =. 4 3 a b If we set B = and set AB = BA, we see that 3a + 4b = 3a 4c, 4a + 3b = 3b 4d,
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More informationMODULES OVER A PID. induces an isomorphism
MODULES OVER A PID A module over a PID is an abelian group that also carries multiplication by a particularly convenient ring of scalars. Indeed, when the scalar ring is the integers, the module is precisely
More informationGalois fields/1. (M3) There is an element 1 (not equal to 0) such that a 1 = a for all a.
Galois fields 1 Fields A field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except by zero) can be performed, and satisfy the usual rules. More
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.15.4, 6.16.2 and 7.17.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationDIVISORS ON NONSINGULAR CURVES
DIVISORS ON NONSINGULAR CURVES BRIAN OSSERMAN We now begin a closer study of the behavior of projective nonsingular curves, and morphisms between them, as well as to projective space. To this end, we introduce
More informationAlgebraic groups Lecture 1
Algebraic groups Lecture Notes by Tobias Magnusson Lecturer: WG September 3, 207 Administration Registration: A sheet of paper (for registration) was passed around. The lecturers will alternate between
More informationRings. Chapter 1. Definition 1.2. A commutative ring R is a ring in which multiplication is commutative. That is, ab = ba for all a, b R.
Chapter 1 Rings We have spent the term studying groups. A group is a set with a binary operation that satisfies certain properties. But many algebraic structures such as R, Z, and Z n come with two binary
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationTHE MINIMAL POLYNOMIAL AND SOME APPLICATIONS
THE MINIMAL POLYNOMIAL AND SOME APPLICATIONS KEITH CONRAD. Introduction The easiest matrices to compute with are the diagonal ones. The sum and product of diagonal matrices can be computed componentwise
More informationSome notes on linear algebra
Some notes on linear algebra Throughout these notes, k denotes a field (often called the scalars in this context). Recall that this means that there are two binary operations on k, denoted + and, that
More informationEXAM 2 REVIEW DAVID SEAL
EXAM 2 REVIEW DAVID SEAL 3. Linear Systems and Matrices 3.2. Matrices and Gaussian Elimination. At this point in the course, you all have had plenty of practice with Gaussian Elimination. Be able to row
More informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
More informationLINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS
LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationChapter 1. WedderburnArtin Theory
1.1. Basic Terminology and Examples 1 Chapter 1. WedderburnArtin Theory Note. Lam states on page 1: Modern ring theory began when J.J.M. Wedderburn proved his celebrated classification theorem for finite
More informationLECTURE 13. Quotient Spaces
LECURE 13 Quotient Spaces In all the development above we have created examples of vector spaces primarily as subspaces of other vector spaces Below we ll provide a construction which starts with a vector
More informationLinear Algebra, Summer 2011, pt. 2
Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................
More information1 Basics of vector space
Linear Algebra Review And Beyond Lecture 1 In this lecture, we will talk about the most basic and important concept of linear algebra vector space. After the basics of vector space, I will introduce dual
More informationMath 210B. Artin Rees and completions
Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an Amodule. In class we defined the Iadic completion of M to be M = lim M/I n M. We will soon show
More informationAng aking kontrata: Ako, si, ay nangangakong magsisipag magaral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking
Ang aking kontrata: Ako, si, ay nangangakong magsisipag magaral hindi lang para sa aking sarili kundi para rin sa aking pamilya, para sa aking bayang Pilipinas at para sa ikauunlad ng mundo. THEOREMS
More informationLinear Algebra II. 2 Matrices. Notes 2 21st October Matrix algebra
MTH6140 Linear Algebra II Notes 2 21st October 2010 2 Matrices You have certainly seen matrices before; indeed, we met some in the first chapter of the notes Here we revise matrix algebra, consider row
More informationLinear Algebra (Math324) Lecture Notes
Linear Algebra (Math324) Lecture Notes Dr. Ali Koam and Dr. Azeem Haider September 24, 2017 c 2017,, Jazan All Rights Reserved 1 Contents 1 Real Vector Spaces 6 2 Subspaces 11 3 Linear Combination and
More informationHomework 5 M 373K Mark Lindberg and Travis Schedler
Homework 5 M 373K Mark Lindberg and Travis Schedler 1. Artin, Chapter 3, Exercise.1. Prove that the numbers of the form a + b, where a and b are rational numbers, form a subfield of C. Let F be the numbers
More informationNONCOMMUTATIVE POLYNOMIAL EQUATIONS. Edward S. Letzter. Introduction
NONCOMMUTATIVE POLYNOMIAL EQUATIONS Edward S Letzter Introduction My aim in these notes is twofold: First, to briefly review some linear algebra Second, to provide you with some new tools and techniques
More informationMathematics Department Stanford University Math 61CM/DM Vector spaces and linear maps
Mathematics Department Stanford University Math 61CM/DM Vector spaces and linear maps We start with the definition of a vector space; you can find this in Section A.8 of the text (over R, but it works
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationCommutative Rings and Fields
Commutative Rings and Fields 1222017 Different algebraic systems are used in linear algebra. The most important are commutative rings with identity and fields. Definition. A ring is a set R with two
More informationWe showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true.
Dimension We showed that adding a vector to a basis produces a linearly dependent set of vectors; more is true. Lemma If a vector space V has a basis B containing n vectors, then any set containing more
More informationA finite universal SAGBI basis for the kernel of a derivation. Osaka Journal of Mathematics. 41(4) P.759P.792
Title Author(s) A finite universal SAGBI basis for the kernel of a derivation Kuroda, Shigeru Citation Osaka Journal of Mathematics. 4(4) P.759P.792 Issue Date 20042 Text Version publisher URL https://doi.org/0.890/838
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationMATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix.
MATH 323 Linear Algebra Lecture 12: Basis of a vector space (continued). Rank and nullity of a matrix. Basis Definition. Let V be a vector space. A linearly independent spanning set for V is called a basis.
More informationLinear Algebra. Paul Yiu. Department of Mathematics Florida Atlantic University. Fall 2011
Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 Linear Algebra Paul Yiu Department of Mathematics Florida Atlantic University Fall 2011 1A: Vector spaces Fields
More informationStructure Theorem for Semisimple Rings: WedderburnArtin
Structure Theorem for Semisimple Rings: WedderburnArtin Ebrahim July 4, 2015 This document is a reorganization of some material from [1], with a view towards forging a direct route to the Wedderburn Artin
More informationINTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS
INTRODUCTION TO REPRESENTATION THEORY AND CHARACTERS HANMING ZHANG Abstract. In this paper, we will first build up a background for representation theory. We will then discuss some interesting topics in
More informationW2 ) = dim(w 1 )+ dim(w 2 ) for any two finite dimensional subspaces W 1, W 2 of V.
MA322 Sathaye Final Preparations Spring 2017 The final MA 322 exams will be given as described in the course web site (following the Registrar s listing. You should check and verify that you do not have
More informationLinear Algebra Highlights
Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to
More informationAbstract Algebra, Second Edition, by John A. Beachy and William D. Blair. Corrections and clarifications
1 Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair Corrections and clarifications Note: Some corrections were made after the first printing of the text. page 9, line 8 For of the
More informationReview 1 Math 321: Linear Algebra Spring 2010
Department of Mathematics and Statistics University of New Mexico Review 1 Math 321: Linear Algebra Spring 2010 This is a review for Midterm 1 that will be on Thursday March 11th, 2010. The main topics
More informationPresentation 1
18.704 Presentation 1 Jesse Selover March 5, 2015 We re going to try to cover a pretty strange result. It might seem unmotivated if I do a bad job, so I m going to try to do my best. The overarching theme
More informationTopics in Module Theory
Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study
More informationStructure of rings. Chapter Algebras
Chapter 5 Structure of rings 5.1 Algebras It is time to introduce the notion of an algebra over a commutative ring. So let R be a commutative ring. An Ralgebra is a ring A (unital as always) together
More information