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Course Question & AnswerAbstract Algebra II
Purpose of Course showclose
This course is a continuation of Abstract Algebra I: we will revisit structures like groups, rings, and fields as well as mappings like homomorphisms and isomorphisms. We will also take a look at ring factorization, which will lead us to a discussion of the solutions of polynomials over abstracted structures instead of numbers sets. We will end the section on rings with a discussion of general lattices, which have both set and logical properties, and a special type of lattice known as Boolean algebra, which plays an important role in probability. We will also visit an important topic in mathematics that you have likely encountered already: vector spaces. Vector spaces are central to the study of linear algebra, but because they are extended groups, group theory and geometric methods can be used to study them.
Later in this course, we will take a look at more advanced topics and consider several useful theorems and counting methods. We will end the course by studying Galois theory—one of the most important theories in algebra, but one that requires a thorough understanding of much of the content we will study beforehand.
Learning Outcomes showclose
- Compute the sizes of finite groups when certain properties are known about those groups.
- Identify and manipulate solvable and nilpotent groups.
- Determine whether a polynomial ring is divisible or not and divide the polynomial (if it is divisible).
- Determine the basis of a vector space, change bases, and manipulate linear transformations.
- Define and use the Fundamental Theorem of Invertible Matrices.
- Use Galois theory to find general solutions of a polynomial over a field.
Course Requirements showclose
√ Have access to a computer
√ Have continuous broadband Internet access
√ Have the ability/permission to install plug-ins or software (e.g. Adobe Reader, Quicktime, Flash viewer)
√ Be competent in the English language
√ Have read the Saylor Student Handbook.
√ Have completed MA211 (Linear Algebra I), MA212 (Linear Algebra II) and MA231 (Abstract Algebra I)
Unit Outline show close
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Unit 1: Advanced Discussion on Groups
We will begin this unit with a series of counting methods involving groups. We will start with Lagrange’s Theorem. Joseph Lagrange discovered the interesting fact that the order, or number of elements, of any true subgroup of a finite group will divide the order of the main group. This is important to know: if a group’s order is prime, then the group is a cyclic, simple group, containing no nontrivial subgroups.
Unit 1 Time Advisory show close
Remembering that a symmetric group is the collection of all permutations of a set X, we can think of a group action as a group homomorphism from some group G to the symmetric group of X if the mapping from G’s identity element to the permutation group of X is the identity transformation and the mapping from gh in G is the composition of permutations assigned to g and h. That is, if we think of X as a set of points on some regular geometric figure, then the group action is a way of describing rotations and reflections of the figure. Group actions are powerful; they allow geometric ideas to be applied to abstract structures.
The Class Equation tells us how many elements are in the center of a finite group G (the center being the set of all elements in G that are commutative under the operation on G) and in the distinct cosets consisting of elements of G not in the center. Remember that by Lagrange’s Theorem, the order of the center must divide the order of G, and if the order of G is prime, then the center is trivial. If the center contains only the identity element, or the center contains all of G, then G is commutative. On the other hand, if G contains non-commutative elements and has a nontrivial center, its order cannot be prime.
Suppose G is a group that acts on a set X. That is, for every g in G and x in X, gx = y in X. The set of all permutations of x caused by left products with g is called the group orbit of x. (Think of x as some coordinate of an object in space and the transformation group G as the group that moves or transforms x around a path.) Burnside’s counting theorem, also called the Cauchy-Frobenius lemma, tells us exactly how many orbits exist for G and X. Burnside did not discover this theorem, but credited Ferdinand Frobenius.
The Sylow theorems involve another counting principle. Peter Sylow said that every finite group has only so many subgroups of a fixed order. His theorem tells us how many subgroups of a given order exist. Further, he said that any subgroup that has a power of a prime order must be the maximal subgroup of that order—that is, such a subgroup cannot be contained in another subgroup of the same order.
Abelian groups are commutative groups. They were first studied by Norwegian mathematician Niels Henrik Abel. Abel discovered that if a group of an equation was commutative, then its roots were solvable by radicals. After studying abelian groups, we will look at solvable groups, which are constructed from abelian groups using extensions. From the use of extensions, a group G is called solvable if its derived series eventually reaches the trivial subgroup of identity. The derived series is a series of commutator subgroups for which commutators are elements of a group G that have the form g-1h-1gh for g, h in G. It makes sense that the only way g-1h-1gh could reduce to the identity element of G, e, is if g and h are commutative and are hence in the center of G. The commutator subgroups are not typically commutative, and the larger that the largest commutator subgroup of G is, the “less abelian” G is.
Nilpotent groups are “almost” abelian through repeated use of commutators. What we discover is that if a group G is a direct product of its Sylow subgroups, it is also nilpotent. If G is nilpotent, then every maximal proper subgroup of G is normal. Nilpotency is an important concept because constructs that are not abelian or solvable may have solvable subconstructs, and there are applications for nilpotent groups in Galois theory.
Unit 1 Learning Outcomes show close
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1.1 Lagrange’s Theorem
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Cosets and Lagrange’s Theorem”
Link: Stephen F. Austin State University: Thomas W. Judson’s "Abstract Algebra Theory and Applications": “Cosets and Lagrange’s Theorem” (PDF)
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Instructions: Please read Chapter 6: Cosets and Lagrange’s Theorem, pages 92 – 97.
Note: This PDF file will be used for the entire course. It will be referenced for readings and assignments throughout.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML).See a broken link? Please let us know!
- Lecture: VideoLecture.net: Marconi Barbosa’s “Group Theory in Machine Learning”
Link: VideoLecture.net: Marconi Barbosa’s Group Theory in Machine Learning (YouTube)
Instructions: Please click on the link above. The video contains information on applications of group theory to the study of machine learning and this clip shows how Lagrange's Theorem may be applied.
Terms of Use: The video is freely available under the Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 license. Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Cosets and Lagrange’s Theorem”
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1.2 Group Actions
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions” (PDF)
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Instructions: Please read Chapter 14: Group Actions, pages 209 – 223.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML).See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 1, 2, and 3”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 1, 2, and 3” (PDF)
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Instructions: Do problems 1, 2, and 3 on page 224. The solutions can be found on page 403.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”
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1.3 The Class Equation
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “The Class Equation”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “The Class Equation” (PDF)
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Instructions: Please read Chapter 14.2: The Class Equation, pages 213 – 215.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Wikipedia's “Order (group theory)”
Link: Wikipedia’s “Order (group theory)” (PDF)
Instructions: Please click on the link above to read the material. The Class Equation is on the page along with other discussions of counting methods.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 6, 8, and 11”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Exercise Problems 6, 8, and 11” (PDF)
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Instructions: Do problems 6, 8, and 11 on page 224. The solutions can be found on page 403.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “The Class Equation”
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1.4 Burnside’s Counting Theorem
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Burnside's Counting Theorem”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Burnside's Counting Theorem” (PDF)
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Instructions: Please read Chapter 14.3: Burnside's Counting Theorem, pages 215 – 222.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Wikipedia's “Burnside's Lemma”
Link: Wikipedia’s “Burnside's Lemma” (PDF)
Instructions: Please click on the link above to read the material.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Group Actions”: “Burnside's Counting Theorem”
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1.5 The Sylow Theorems
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Sylow Theorems”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Sylow Theorems” (PDF)
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Instructions: Please read Chapter 15: The Sylow Theorems, pages 227 – 235.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Lecture: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Alternating group structure”
Link: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Alternating group structure” (Flash, QuickTime or Audio mp3)
Instructions: Please click on the link, then scroll down to Week 8, Lecture 1. Choose the format most appropriate for your internet connection. Watch the entire video. This video clip discusses alternating group structures, but begins with a discussion on the Sylow theorems.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Sylow Theorems”: “Exercise Problems 1, 2, and 17”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Sylow Theorems”: “Exercise Problems 1, 2, and 17” (PDF)
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Instructions: Do problems 1, 2, and 17 on page 235 – 236. The solutions can be found on page 404.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Sylow Theorems”
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1.6 Abelian Groups
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Finite Abelian Groups”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Finite Abelian Groups” (PDF)
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Instructions: Please read 13.1: Finite Abelian Groups, pages 196 – 201.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Wikipedia's “Abelian Group”
Link: Wikipedia’s “Abelian Group” (PDF)
Instructions: Please click on the link above and read the material.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).See a broken link? Please let us know!
- Lecture: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Review, Kernels, Normality”
Link: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Review, Kernels, Normality” (Flash, QuickTime, or Audio mp3)
Instructions: Please click on the link, then scroll down to Week 2, Lecture 1. Choose the format most appropriate for your internet connection. Watch entire video.
Note on the Media: The video clip discusses kernels, normalities and centers.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 1, 4, and 7”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 1, 4, and 7” (PDF)
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Instructions: Do problems 1, 4, and 7 on page 205 – 206. The solutions can be found on page 403.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Finite Abelian Groups”
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1.7 Solvable Groups
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 12, 16, and 21”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 12, 16, and 21” (PDF)
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Instructions: Do problems 12, 16, and 21 on pages 206 – 207. The solutions can be found on page 403.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Solvable Groups”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Solvable Groups” (PDF)
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Instructions: Please read 13.2: Solvable Groups, pages 201 – 205.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Wikipedia's “Solvable Group”
Link: Wikipedia “Solvable Group” (PDF)
Instructions: Please click on the link above and read the material.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “The Structure of Groups”: “Exercise Problems 12, 16, and 21”
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1.8 Nilpotent Groups
- Reading: Wolfram MathWorld’s “Nilpotent Group”
Link: Wolfram MathWorld's: “Nilpotent Group” (HTML)
Instructions: Please click on the link to read the information on the page. This webpage contains a concise rendering of information on nilpotent groups. It also contains links to related information.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.See a broken link? Please let us know!
- Reading: Wikipedia's “Nilpotent Group”
Link: Wikipeda’s “Nilpotent Group” (PDF)
Instructions: Please click on the link above to read the material. This webpage contains useful information about nilpotent groups.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).See a broken link? Please let us know!
- Reading: Wolfram MathWorld’s “Nilpotent Group”
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Unit 2: Advanced Discussion of Rings
We begin Unit 2 by returning to the idea of polynomial rings, which we first encountered in Abstract Algebra I. These are rings that have a set of polynomials with coefficients in another ring. We will revisit the materials we have seen on these rings first. Then, we will look at the division algorithm. For polynomial rings, this is analogous to the algorithm for dividing integers. Put simply, if there exist polynomials g, q and f in R[X] such that f = gq, then if g|f, one of the following must be true: either f = 0 or deg f ³deg g. Also, if fg = 0, then either f = 0 or g = 0 (or both). What we will find is a procedure for dividing f.
Unit 2 Time Advisory show close
Irreducible polynomials are similar to prime integers, for they cannot be split into the product of two or more non-trivial polynomials. Irreducible polynomials play an important role in Galois theory, because there exists a relationship between a field, the field’s Galois group, and the field’s irreducible polynomials. Since all fields are commutative rings, we will be able to incorporate what we learn in this unit in Unit 4.
Integral domains advance our conversation further, because these bring us closer to the study of fields. Integral domains are commutative rings with the property that the multiplicative and additive identities cannot be equal; they also have no zero divisors. That is, if ab is in the ring and ab = 0, then either a = 0 or b = 0. Both cannot be nonzero. Thus, integral domains are either prime or factorable. If these factors are unique, then the integral domain is a unique factorization domain.
Lattices are algebraic structures with group-like properties. With the operations join Ú (which produces the least upper bound of two elements in the lattice) and meet Ù (which produces the greatest lower bound between two elements on the lattice), lattices are close to being commutative rings. Boolean algebras, then, are complemented distributed lattices. That is, for every a in lattice B, there is an element b for which a Úb = 1 and a Ùb = 0 and the properties Úand Ùdistribute over each other. The complement b of a listed above need not be unique. The importance of Boolean algebras is that they form ring-like structures that use essential properties of both set and logic operations.
Unit 2 Learning Outcomes show close
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2.1 Polynomial Rings
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials” (PDF)
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Instructions: Please read 17.1: Polynomial Rings, pages 264 – 267.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”: “Exercise Problem 2”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Polynomials”: “Exercise Problem 2”
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Instructions: Do problem 2 on page 278. The solution can be found on page 405.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”
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2.2 The Division Algorithm
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Polynomials”
Also available in:
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Instructions: Please read 17.2: The Division Algorithm, pages 268 – 271.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”: “Exercise Problems 3 and 5”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Polynomials”: “Exercise Problems 3 and 5”
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Instructions: Do problems 3 and 5 on page 278 – 279. The solutions can be found on page 405.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”
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2.3 Irreducible Polynomials
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Polynomials”
Also available in:
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Instructions: Please read 17.3: Irreducible Polynomials, pages 272 – 277.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF)
Also available in:
iBooks
Instructions: Do problems 3 and 5 on page 278 – 279. The solutions can be found on page 405.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Polynomials”
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2.4 Integral Domains
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Integral Domains”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Integral Domains”
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Instructions: Please read 18.1: Field of Fractions, pages 283 – 287.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Integral Domains”: “Exercise Problem 1”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Integral Domains”: “Exercise Problem 1”
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Instructions: Do problem 1 on page 297. The solution can be found on page 406.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Integral Domains”
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2.5 Factorization of Integral Domains
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Integral Domains”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Integral Domains”
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Instructions: Please read 18.2: Factorization of Integral Domains, pages 287 – 296.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Integral Domains”: “Exercise Problem 2”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Integral Domains”: “Exercise Problem 2”
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Instructions: Do problem 2 on page 297. The solution can be found on page 406.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Integral Domains”
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2.6 Lattices
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Lattices and Boolean Algebras”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Lattices and Boolean Algebras”
Also available in:
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Instructions: Please read 19.1: Lattices, pages 301 – 306.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Lecture: YouTube: “Lecture 40 – Lattices”
Link: YouTube: “Lecture 40 - Lattices” (YouTube)
Instructions: Please click on the link and view the video in its entirety.
Note on the Media: The video discusses linear independence. Professor Kamala Krithvisian, from the Department of Computer Science and Engineering of IIT Madris, in India, created this video. She does an excellent job of discussing lattices in terms of posets and makes the material understandable.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Lattices and Boolean Algebras”: “Exercise Problem 2”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Lattices and Boolean Algebras”: “Exercise Problem 2”
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Instructions: Do problem 2 on page 315. The solution can be found on page 406.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Lattices and Boolean Algebras”
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2.7 Boolean Algebras
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Lattices and Boolean Algebras”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Lattices and Boolean Algebras”
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Instructions: Please read 19.2: Boolean Algebras, pages 306 – 312.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Lattices and Boolean Algebras”: “Exercise Problem 6”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Lattices and Boolean Algebras”: “Exercise Problem 6”
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Instructions: Do problem 6 on page 316. The solution can be found on page 406.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Lattices and Boolean Algebras”
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Unit 3: Vector Spaces
Vector spaces are among the most useful structures in mathematics. Used heavily in economics and finance as well as engineering and the natural and physical sciences, vector spaces are additional structures that have both algebraic and geometric properties.
Unit 3 Time Advisory show close
Vectors are extended commutative groups with additional distributive properties concerning field values called scalars. Thus, all theorems that apply to groups may apply to vectors.
A subspace is a subset of a vector space that contains the zero vector and is a vector space itself. Linear independence is a property of spaces and subspaces that states that no family of vectors in the space may be written as linear combinations of the other vectors in the family. The number of unique, linearly independent vectors in a space is called the space’s dimension.
The basis of a space is a set of vectors that can represent all the vectors in the space by linear combinations. That is, the basis is a linearly independent spanning set of the space. Sometimes, it is easier to work in some bases than others. For that reason, we sometimes prefer to change a basis from one coordinate set to another. The group isomorphism that maps one basis to another is called a change of basis. This type of isomorphism is a category of a set of functions called linear transformations. In general, linear transformations are functions that preserve the operations of vector addition and scalar multiplication. We will discover that all compositions of transformations result in transformations. The kernel of a linear transformation L is the set of all vectors v in a space V for which L(v) = 0. That is, all of the vectors that are mapped to the zero vector by L are in L’s kernel. The kernel of L is by nature a subspace of the vector space V. If the only vector in V contained in the kernel of L (also called Ker(L)) is 0, then L is 1-1. The range of a transformation L is all vectors w in space W for which there is a v in space V such that L(v) = w. If L is onto W, then the range of L = W. If Ker(L) = {0} and range L = W, then L is a vector space isomorphism. If L is an isomorphism, then matrices made from vectors in V are invertible.
At the end of the unit, we will consider the Fundamental Theorem of Invertible Matrices, which is the core theorem of linear algebra. The beauty of this theorem is that there are twenty equivalent statements about matrices. If we determine that any of the twenty are true about a matrix, they are all true. Conversely, if any is not true, none are true.
Unit 3 Learning Outcomes show close
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3.1 Definitions and Examples of Vector Spaces
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”
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Instructions: Please read 20.1: Definitions and Examples, pages 319 – 321.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Lecture: Khan Academy’s “Linear Algebra: Vector Examples”
Link: Khan Academy’s “Linear Algebra: Vector Examples” (YouTube)
Instructions: Please watch the entire lecture, which provides specific examples of vectors on a coordinate plane.
Watching this lecture should take approximately 30 minutes.
Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Khan Academy.See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”: “Exercise Problems 3 and 5”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”: “Exercise Problems 3 and 5”
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Instructions: Do problems 3 and 5 on page 325. The solution can be found on page 407.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”
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3.2 Subspaces
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”
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Instructions: Please read 20.2: Subspaces, pages 321 – 322.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Lecture: Khan Academy’s “Linear Subspaces”
Link: Khan Academy’s “Linear Subspaces”(YouTube)
Instructions: Please watch the entire lecture, which is about linear subspaces.
Watching this lecture should take approximately 30 minutes.
Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Khan Academy.See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”: “Exercise Problem 7”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”: “Exercise Problem 7”
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Instructions: Do problem 7 on page 325. The solution can be found on page 407.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”
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3.3 Linear Independence
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Linear Independence”
Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Linear Independence”
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Instructions: Please read Chapter 2, II: Linear Independence, pages 99 – 106.
Notes on the Textbook: This PDF file will be used for the rest of the unit. It will be referenced for readings and assignments throughout.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”
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Instructions: Please read 20.3: Linear Independence, pages 322 – 325.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Lecture: Khan Academy’s “Introduction to Linear Independence”
Link: Khan Academy’s “Introduction to Linear Independence” (YouTube)
Instructions: Please watch the entire lecture, which is about linear independence.
Watching this lecture should take approximately 15 minutes.
Terms of Use: The video above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Khan Academy.See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Vector Spaces”: “Exercise Problem 15”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Vector Spaces”: “Exercise Problem 15”
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Instructions: Do problem 15 on pages 326 – 327. The solution can be found on page 407.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Linear Independence”: “Exercise Problems 1.18, 1.19, and 1.24”
Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Linear Independence”: “Exercise Problems 1.18, 1.19, and 1.24”
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Instructions: Do problems 1.18, 1.19, and 1.24 on pages 106 – 107. The answers can be found here on pages 49 – 50.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Linear Independence”
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3.4 Change of Basis
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Change of Basis”
Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Change of Basis”
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Instructions: Please read Chapter 3, V: Change of Basis, pages 236 – 245.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Lecture: Khan Academy’s “Linear Algebra: Change of Basis Matrix”
Link: Khan Academy’s “Linear Algebra: Change of Basis Matrix” (YouTube)
Instructions: Please watch the entire lecture, which is about the change of basis matrix.
Watching this lecture should take approximately 20 minutes.
Terms of Use: The video above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Khan Academy.See a broken link? Please let us know!
- Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Exercise Problems 1.6, 1.7, 1.8, and 1.9”
Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Exercise Problems 1.6, 1.7, 1.8, and 1.9”
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Instructions: Please do problems 1.6, 1.7, 1.8, and 1.9 on page 239. The solutions to these exercises can be found here on pages 123 – 124.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Change of Basis”
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3.5 Linear Transformations
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Computing Linear Maps”
Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Computing Linear Maps”
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Instructions: Please read Chapter 3, III: Computing Linear Maps, pages 193 – 203. This subchapter shows linear transformations as matrix operations and mappings. It shows that the two are interchangeable.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Lecture: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Bases and Vectorspaces”
Link: Harvard University Extension: Dr. Benedict Gross’ “Math E-222 Abstract Algebra”: “Bases and Vectorspaces” (Flash, QuickTime, or Audio mp3)
Instructions: Please click on the link and then scroll down to Week 4, Lecture 1. Choose the format most appropriate for your internet connection. Watch the entire video.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Vector Spaces”: “Exercise Problems 1.14, 1.15, and 1.17”
Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Vector Spaces”: “Exercise Problems 1.14, 1.15, and 1.17”
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Instructions: Do problems 1.14, 1.15, and 1.17 on page 201. The solutions can be found here on pages 97 – 98.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Computing Linear Maps”
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3.6 Composition of Transformations
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Matrix Multiplication”
Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Matrix Multiplication”
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Instructions: Please read Chapter 3, IV.2: Matrix Multiplication, pages 213 – 220. This subchapter demonstrates composition of linear transformations as matrix multiplication.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Lecture: Khan Academy’s “Compositions of Linear Transformation 1” and “Compositions of Linear Transformation 2”
Link: Khan Academy’s “Compositions of Linear Transformations 1” and “Compositions of Linear Transformations 2” (YouTube)
Instructions: Please watch both lectures, which cover compositions of linear transformations.
Watching these lectures should take approximately 30 minutes.
Terms of Use: The videos above are released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). They attributed to Khan Academy.See a broken link? Please let us know!
- Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Matrix Multiplication”: “Exercise Problems 2.24, 2.26, and 2.29”
Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Matrix Multiplication”: “Exercise Problems 2.24, 2.26, and 2.29”
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Instructions: Do problems 2.24, 2.26, and 2.29 on page 218 – 219. The solutions can be found here on pages 322 – 323.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Matrix Multiplication”
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3.7 Kernel and Range of Transformations
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Rangespace and Nullspace”
Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Rangespace and Nullspace”
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Instructions: Please read Chapter 3, II.2: Rangespace and Nullspace, pages 181 – 192.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Lecture: Khan Academy’s “Preimage and Kernel Example”
Link: Khan Academy’s “Preimage and Kernel Example” (YouTube)
Instructions: Please watch the lecture, which discusses kernel and preimages for vectors in the range of the transformation.
Watching this lecture should take approximately 15 minutes.
Terms of Use: The article above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to the Khan Academy.See a broken link? Please let us know!
- Lecture: Khan Academy’s “im(T): Image of a Transformation”
Link: Khan Academy’s “im(T): Image of a Transformation”(YouTube)
Instructions: Please watch the lecture, which discusses image (or range) of a transformation.
Watching this lecture should take approximately 15 minutes.
Terms of Use: The video above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Khan Academy.See a broken link? Please let us know!
- Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Homomorphisms”: “Rangespace and Nullspace”: “Exercise Problems 2.22, 2.23, 2.24, 2.27, and 2.29”
Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Homomorphisms”: “Rangespace and Nullspace”: “Exercise Problems 2.22, 2.23, 2.24, 2.27, and 2.29”
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Instructions: Do problems 2.22, 2.23, 2.24, 2.27, and 2.29 on page 190. The solutions can be found here on pages 303-304.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Rangespace and Nullspace”
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3.8 Vector Space Isomorphisms
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Isomorphisms”
Link: St. Michael's College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Isomorphisms”
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Instructions: Please read Chapter 3, I : Isomorphisms, pages 157 – 166.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Assessment: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Isomorphisms”: “Exercise Problems 1.10, 1.11, 1,13, and 1.14”
Link: St. Michael’s College: Jim Hefferon’s “Linear Algebra” (PDF): “Maps Between Spaces”: “Isomorphisms”: “Exercise Problems 1.10, 1.11, 1,13, and 1.14”
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Instructions: Do problems 1.10, 1.11, 1.13, and 1.14 on page 164. The answers can be found here on pages 287-290.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages iv – vi. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Jim Hefferon and the original version can be found here (HTML).See a broken link? Please let us know!
- Reading: St. Michael's College: Jim Hefferon’s “Linear Algebra”: “Maps Between Spaces”: “Isomorphisms”
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3.9 The Fundamental Theorem of Invertible Matrices
- Reading: Wikipedia's “Invertible Matrix”
Link: Wikipedia’s “Invertible Matrix” (PDF)
Instructions: This article contains useful information about invertible matrices, including the twenty equivalent statements contained in the Fundamental Theorem of Invertible Matrices, listed as properties of invertible matrices.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).See a broken link? Please let us know!
- Lecture: Khan Academy’s “Linear Algebra: Simplifying Conditions For Invertibility”
Link: Khan Academy’s “Linear Algebra: Simplifying Conditions For Invertibility” (YouTube)
Instructions: Please watch the entire video, which discusses conditions for invertibility.
Watching this video should take approximately 10 minutes.
Terms of Use: The video above is released under Creative Commons Attribution-NonCommercial-ShareAlike 3.0 (HTML). It is attributed to Khan Academy.See a broken link? Please let us know!
- Reading: Wikipedia's “Invertible Matrix”
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Unit 4: Galois Theory
Unit 4 concerns one of the most important theories in abstract algebra: Galois theory. Evariste Galois is one of the most enigmatic figures in mathematics. Incredibly intelligent, he intimidated many of the finest mathematical minds in France as a teenager. A proponent of democracy at the height of monarchy in France, the hotheaded Galois was tricked into accepting a duel against a veteran swordsman, Pescheux d’Herbinville, at age 20. Realizing that he probably would not survive the duel, Galois spent much of his last night alive frantically compiling ideas he had on general solutions of polynomials. The notes he wrote to his friend Auguste Chevalier became the foundation for what we now know as Galois theory.
Unit 4 Time Advisory show close
First, Galois used permutation groups to show how roots of a polynomial are related. After learning about permutation groups, we will look at Galois groups, which are sets of automorphisms on an extension field L of some field K. We will then conclude the unit (and the course) with a long look at the Fundamental Theorem of Galois Theory, which states that subgroups of the Galois group for extension field L for some field K correspond with the subfields of K that contain L.
Unit 4 Learning Outcomes show close
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4.1 Permutation Groups and Solutions of Polynomials
- Lecture: YouTube: “Symmetries of a Star and its Permutation Group”
Link: YouTube: “Symmetries of Star and its Permutation Group” (YouTube)
Instructions: Please click on the link and view the video in its entirety. The video discusses a permutation group of a regular geometric figure (a Star of David). You first viewed this interesting visual presentation in Abstract Algebra I. Once again, this video may be viewed in a new light due to the material we have covered up to this point.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Reading: Wolfram MathWorld: “Permutation Group”
Link: Wolfram MathWorld: “Permutation Group” (HTML)
Instructions: Please click on the link to read the information on the page. This webpage contains a concise rendering of information on permutation groups. It is short, but provides a good explanation in a few sentences. The page also contains a set of useful links to related topics.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.See a broken link? Please let us know!
- Reading: Wikipedia’s “Permutations”
Link: Wikipedia’s “Permutation Group” (PDF)
Instructions: Please read the entire article. This webpage contains a concise rendering of information on permutation groups, including various examples.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML). You can find the original Wikipedia version of this article here (HTML).See a broken link? Please let us know!
- Lecture: YouTube: “Group Theory Permutations”
Link: YouTube: “Group Theory Permutations” (YouTube)
Instructions: Please click on the link and view the video in its entirety. The video discusses general permutations and then discusses permutations in light of groups. This is a good introduction to the concept. We saw this video in Abstract Algebra I, but in light of all we have covered since, you may view this video in a new light.
Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.See a broken link? Please let us know!
- Lecture: YouTube: “Symmetries of a Star and its Permutation Group”
-
4.2 Galois Groups
- Reading: Wolfram MathWorld: “Galois Group”
Link: Wolfram MathWorld: “Galois Group” (HTML)
Instructions: Please click on the link to read the information on the page. This webpage contains a concise rendering of information on Galois groups. It is short, but provides a solid explanation in just a few sentences. The page also contains a set of useful links to related topics.
Terms of Use: Please respect Wolfram MathWorld's terms of use. MathWorld webpages are free for academic use and may be hyperlinked, according to their FAQ site.See a broken link? Please let us know!
- Reading: Wikipedia's “Galois Group”
Link: Wikipedia’s “Galois Group” (HTML)
Instructions: Please click on the link above to read the material. This webpage contains useful information about the Galois Group.
Terms of Use: The article above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).See a broken link? Please let us know!
- Reading: Wolfram MathWorld: “Galois Group”
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4.3 The Fundamental Theorem of Galois Theory
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Galois Theory”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Galois Theory”
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Instructions: Please read 23.2: The Fundamental Theorem, pages 377 – 383.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Assessment: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Galois Theory”: “Exercise Problems 1, 2, 3, and 9”
Link: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications” (PDF): “Galois Theory”: “Exercise Problems 1, 2, 3, and 9”
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Instructions: Do problems 1, 2, 3, and 9 on page 391. The solution can be found on page 409.
Terms of Use: Please respect the copyright, license, and terms of use displayed on pages 410 – 417. The material linked above is licensed under the GNU Free Documentation License (HTML). It is attributed to Thomas W. Judson and the original version can be found here (HTML)See a broken link? Please let us know!
- Reading: Stephen F. Austin State University: Thomas W. Judson’s “Abstract Algebra Theory and Applications”: “Galois Theory”
-
Final Exam
- Final Exam: The Saylor Foundation's MA232 Final Exam
Link: The Saylor Foundation's MA232 Final Exam
Instructions: You must be logged into your Saylor Foundation School account in order to access this exam. If you do not yet have an account, you will be able to create one, free of charge, after clicking the link.See a broken link? Please let us know!
- Final Exam: The Saylor Foundation's MA232 Final Exam
