Covariance

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Covariance

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00-09 Atlas

🗺️ Map of Content - Algebra Qual

🗺️ Map of Content - Algebra Theory

🗺️ Map of Content - Category Theory

🗺️ Map of Content - Modules

🗺️ Map of Content - Ring Theory

10-19 Teaching

11 Classes

MATH 561 - Graduate Algebra

2024 - Fall

Homework

Study Guides

Study Guide for Final Exam

Study Guide for Midterm Exam

Daily class summaries

MATH 561 Home

Midterm Exam Solutions

Exercise Solutions

Annihilators - Solution

Exercises

A criterion for a module to be finitely generated

Alternate characterization of rank

An intramural isomorphism

Annihilators of torsion modules

Annihilators

Cokernels in the category of abelian groups

Direct sum of free modules

Direct sums and injective, projective, flat

Equalizers and coequalizers of matrices

Equivalence relations on sets

Finite abelian groups are neither injective nor projective

Group morphisms that cannot define module morphisms

Primes and annihilators

Quotients of cyclic modules

Rank and quotients

Rational and Jordan canonical forms

Ring property from module property

Submodules via ideals

Tensor product of projective modules is projective

Tensor product of rings is a coproduct

The evaluation map

The pullback functor is an adjoint

The Short Five Lemma

There is no 'center' functor on Grp

Torsion Z-modules

Torsion submodules

14 Algebra Qual

Previous exams

Problem bank

Pool problems

Group theory

A condition to be non-cyclic

A condition under which a group must be abelian (2)

A dihedral group that is not an internal direct product

A group isomorphic to a subgroup of a direct product of quotient groups

A group isomorphic to an internal direct product

A group of upper-triangular matrices

A group with a trivial automorphism group (2)

A group with a trivial automorphism group

A property of the order of an element

An automorphism of a group of odd order

Another condition for a group to be abelian

Automorphisms of a finite cyclic group

Comparing cosets

Computations with inner automorphisms

Elements of finite order

Elements of order 2

Existence of a normal subgroup of finite index

Existence of an identity element in a group

Existence of automorphisms

Image of a normal subgroup and induced morphisms

Image of the identity is the identity

Indices and intersections

Inner automorphisms and the center of a group

Nonexistence of a simple group of a given order

Nonexistence of small nonabelian groups

Normal subgroups with trivial intersection

Normality and the operation on cosets (defunct)

Normality and the operation on cosets

Normalizer of a subgroup

Normalizers and centralizers

Order of a power of an element

Order of an element in a finite group

Orders of elements in a quotient group

Preimage of a subgroup

Product of two subgroups

Products of quotient groups

Projection onto a quotient

Properties of the center of a group

Property of the order of an element

Proving Lagrange's Theorem

Stabilizer of a coset

Subgroups of a group of even order

The integers as a subgroup of the rationals

The Third Isomorphism Theorem

Linear algebra

A condition ensuring diagonalizability

A property of surjective linear transformations

An upper bound on the number of nonzero eigenvalues

Analyzing an unusual matrix

Direct sums and idempotent transformations

Dot product and cross product as linear transformations

Eigenvalues and eigenspaces of a matrix with a given property

Eigenvectors and linear independence

Eigenvectors of commuting linear transformations

Eigenvectors with distinct eigenvalues are linearly independent

Linear dependence and relations

Linear endomorphism of a vector space of matrices

Matrix representing a linear transformation

Numerical range of a linear transformation

Projections and adjoints

Properties of transpose

Skew-symmetric matrices

Sum and union of subspaces

Ring theory

A non-PID

A ring in which all prime ideals are maximal

Automorphisms of a ring

Boolean algebras

Boolean rings are commutative

Characteristic of a ring (2)

Characteristic of a ring

Dimension of a PID

Euclidean domains are PIDs

Existence of an identity element in a finite ring

Generator for a field extension

Ideals in a polynomial ring (2)

Ideals in a polynomial ring

Idempotent elements in a ring (2)

Idempotent elements in a ring

Image of the identity element under a ring morphism

Maximal ideals in a PID

Nilpotent elements in a ring

Nilpotent elements

Nonzero prime ideals are maximal in a PID

Polynomials with even constant term

Prime and irreducible elements in a commutative ring

Prime ideals and quotient rings

Properties of Boolean rings

Properties of the annihilator

Proving an ideal is prime

Quotienting out nilpotent elements

Quotients and direct products

The nilradical of a ring

The structure of the integers as both a group and a ring

Pool problems in group theory

Pool problems in linear algebra

Pool problems in ring theory

Template problems

Group theory

A condition under which a group must be abelian

Computations in symmetric groups

Computing an automorphism group

Counting morphisms between specified groups

Directly proving the existence of an element of a desired order

Finding all morphisms between two groups

Inner and outer automorphisms

Inner automorphisms of an alternating group

Nonexistence of morphisms between two groups

Order of elements in a symmetric group

The cyclic group of order 2020

Verifying axioms of a group

Working modulo 11

Linear algebra

A linear transformation from a vector space of polynomials

A vector space of square matrices

Bases for a subspace and its orthogonal complement

Computations with a given linear transformation

Diagonalization and matrix powers

Diagonalization of a given matrix (2)

Diagonalization of a given matrix

Dimension of a subspace and its orthogonal complement

Jordan canonical form of a matrix (2)

Jordan canonical form of a matrix

Linear transformation from a vector space of polynomials

Matrix and characteristic polynomial for a given linear transformation

Matrix and eigenvalues of a given linear transformation

Orthogonal complements

Orthogonal projection and scaling

Orthogonal projection onto a line (2)

Orthogonal projection onto a line (3)

Orthogonal projection onto a line

Orthogonal projection onto a plane (2)

Orthogonal projection onto a plane

Projection onto a plane

Radial expansion from a fixed line

Reflection across a plane

Rotation around an axis

Ring theory

A maximal ideal in a function ring

An evaluation morphism

An isomorphism of rings

Constructing a field extension

Constructing the field with eight elements

Evaluation at i

Existence of certain ring morphisms

Group of units of a product

Image of an evaluation morphism

Morphism from the Gaussian integers (2)

Morphism from the Gaussian integers

The field with eight elements

The field with nine elements

The kernel of an evaluation morphism

Using the Chinese Remainder Theorem

Template problems in group theory

Template problems in linear algebra

Template problems in ring theory

20-29 Research

24 Summer REUs

2024

Summer REU 2024

40-49 Knowledge

41 Mathematics

Algebra theory

Algebras

Exterior algebras

Symmetric algebras

Symmetric and alternating tensors

Tensor algebras

Algebraic geometry

Examples of classical conic duality

Calculus

Determining and classifying local extrema

Determining constrained extrema

Category theory

Abelian Categories

Abelian categories

Additive categories

Chain complexes

Diagram chases without elements

Diagram lemmas

Double complexes and mural maps

Exact sequences and chain homology

Preadditive categories

Spectral sequences

The Salamander Lemma

Adjoints

Examples of adjoints

Properties of adjoints

Basic Structures

Direct sum of free modules

Prove that any direct sum of free $R$ -modules is free.

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