Канал: Adam Glesser

Between Two Chairs

Between Two Chairs

The determinant via permutations

The determinant via permutations

The determinant as a product of eigenvalues

The determinant as a product of eigenvalues

The trace is basis-invariant

The trace is basis-invariant

The trace and the eigenvalues

The trace and the eigenvalues

Properties of the Riesz element

Properties of the Riesz element

Stating and applying the complex spectral theorem

Stating and applying the complex spectral theorem

A normal operator that is not self-adjoint

A normal operator that is not self-adjoint

An abstract explicit isomorphism between vector spaces

An abstract explicit isomorphism between vector spaces

The trace

The trace

A property of the image of a self-adjoint operator

A property of the image of a self-adjoint operator

Determining self-adjointness from the matrix

Determining self-adjointness from the matrix

The adjoint of a composition

The adjoint of a composition

Finding the image of the adjoint map

Finding the image of the adjoint map

Invertible diagonal matrix with indistinct diagonal entries

Invertible diagonal matrix with indistinct diagonal entries

Finding the matrix of the operator given an eigenvalue

Finding the matrix of the operator given an eigenvalue

Explicitly finding the orthogonal complement

Explicitly finding the orthogonal complement

Orthogonal complements reverse subset containment

Orthogonal complements reverse subset containment

Proof of the Existence of Riesz elements

Proof of the Existence of Riesz elements

Applying the (modified) Gram–Schmidt method

Applying the (modified) Gram–Schmidt method

Upper-triangular with respect to an orthonormal basis

Upper-triangular with respect to an orthonormal basis

Finding eigenvalues and eigenvectors at the same time

Finding eigenvalues and eigenvectors at the same time

Pulling a scalar out of a norm

Pulling a scalar out of a norm

Orthogonality of the orthogonal decomposition

Orthogonality of the orthogonal decomposition

Verifying the inner-product axioms

Verifying the inner-product axioms

The transpose is a linear isomorphism

The transpose is a linear isomorphism

Enough distinct eigenvalues yields an eigenbasis

Enough distinct eigenvalues yields an eigenbasis

The significance of a diagonal matrix

The significance of a diagonal matrix

Properties of upper triangular matrices

Properties of upper triangular matrices

Range of a composition

Range of a composition

Another null space problem

Another null space problem

Eigenvectors and one-dimensional invariant subspaces

Eigenvectors and one-dimensional invariant subspaces

Finding eigenstuff for a projection

Finding eigenstuff for a projection

Extending a linearly independent list to a basis

Extending a linearly independent list to a basis

Basis and Dimension: Truth or Dare

Basis and Dimension: Truth or Dare

An explicit isomorphism between a vector space of dimension n and Fⁿ.

An explicit isomorphism between a vector space of dimension n and Fⁿ.

Using the matrix to compute the map

Using the matrix to compute the map

Another span begets span problem

Another span begets span problem

A vector space axiom fail

A vector space axiom fail

State and apply the fundamental theorem of linear maps

State and apply the fundamental theorem of linear maps

Finding a basis for the range

Finding a basis for the range

A trivial null space implies injectivity

A trivial null space implies injectivity

Is it a linear map?

Is it a linear map?

A long enough linearly independent list is a basis

A long enough linearly independent list is a basis

Bases for ℂ

Bases for ℂ

Function Terminology

Function Terminology

A linearly independent list begets a linearly independent list

A linearly independent list begets a linearly independent list

Dependent or independent?

Dependent or independent?

Correcting an error in the definition of linear independence

Correcting an error in the definition of linear independence

A spanning list begets a spanning list

A spanning list begets a spanning list

A list of vectors that does not span ℝ²

A list of vectors that does not span ℝ²

The span of three abstract vectors

The span of three abstract vectors

Set-builder notation

Set-builder notation

Determining if a subset is a subspace

Determining if a subset is a subspace

Vector space axioms and consequences

Vector space axioms and consequences

Is it a Group?

Is it a Group?

Identity in a Group is Unique

Identity in a Group is Unique

Using Lagrange's Theorem to Determine Possible Subgroup Orders

Using Lagrange's Theorem to Determine Possible Subgroup Orders

Subgroups of S₄?

Subgroups of S₄?

A MAD Hack

A MAD Hack

Basics of Quotient Rings

Basics of Quotient Rings

Units, Zero Divisors, Inverses, and Annihilators

Units, Zero Divisors, Inverses, and Annihilators

Units of a Polynomial Ring over an Integral Domain

Units of a Polynomial Ring over an Integral Domain

Quotient by Ideal Generated by Indecomposable Element is a Field

Quotient by Ideal Generated by Indecomposable Element is a Field

Another Unique Factorization Fail

Another Unique Factorization Fail

Basic Properties of Integral Domains

Basic Properties of Integral Domains

Computing a GCD of Two Polynomials

Computing a GCD of Two Polynomials

The Additive Identity Multiplicatively Annihilates Every Element

The Additive Identity Multiplicatively Annihilates Every Element

A Clever Carl Proof that Violates Lagrange's Theorem

A Clever Carl Proof that Violates Lagrange's Theorem

Deciding Whether Subsets are Subgroups

Deciding Whether Subsets are Subgroups

Reviewing the Group Axioms

Reviewing the Group Axioms

Proving the Cancellation Law for Groups

Proving the Cancellation Law for Groups

Debunking Some Misconceptions about Equivalence Relations

Debunking Some Misconceptions about Equivalence Relations

A Non-Principal Ideal of ℤ[x]

A Non-Principal Ideal of ℤ[x]

A Relation on ℤ that is Not an Equivalence Relation

A Relation on ℤ that is Not an Equivalence Relation

Ideal Properties That May or May Not Be True

Ideal Properties That May or May Not Be True

The Kernel of a Ring Homomorphism is an Ideal

The Kernel of a Ring Homomorphism is an Ideal

Some Quotient Rings Basics

Some Quotient Rings Basics

Prime and Maximal Ideals

Prime and Maximal Ideals

Divisibility and the GCD of Ideals

Divisibility and the GCD of Ideals

The Intersection of Ideals is an Ideal

The Intersection of Ideals is an Ideal

Is this Subset of Matrices an Ideal?

Is this Subset of Matrices an Ideal?

Exponentials, Logs, and a Hint of Lambert

Exponentials, Logs, and a Hint of Lambert

Indecomposable Factorization of 8 in the "Glesser Integers"

Indecomposable Factorization of 8 in the "Glesser Integers"

Finding the 2-Part of a Positive Integer

Finding the 2-Part of a Positive Integer

Prime Power Division

Prime Power Division

Gaussian Integers with Norm 1 are Units

Gaussian Integers with Norm 1 are Units

Applying the Division Algorithm for Gaussian Integers

Applying the Division Algorithm for Gaussian Integers

What is True about F[x]?

What is True about F[x]?

Transversals and Quotient Sets

Transversals and Quotient Sets

The Remainder When Dividing by x-a

The Remainder When Dividing by x-a

Units, Zero Divisors, and Inverses in ℤ/24ℤ

Units, Zero Divisors, and Inverses in ℤ/24ℤ

Ring Terminology Basics and Examples

Ring Terminology Basics and Examples

Computing the Order Modulo 10

Computing the Order Modulo 10

Modular Arithmetic in 2023

Modular Arithmetic in 2023

2023 as a Sum of Squares?

2023 as a Sum of Squares?

Examples and Nonexamples of Functions

Examples and Nonexamples of Functions

Irrationality and the Fundamental Theorem of Arithmetic

Irrationality and the Fundamental Theorem of Arithmetic

Euclid's Lemma and a Counterexample

Euclid's Lemma and a Counterexample

The Extended Euclidean Algorithm

The Extended Euclidean Algorithm

Bézout's Lemma Subtleties

Bézout's Lemma Subtleties

The Euclidean Algorithm Applied

The Euclidean Algorithm Applied

The GCD Divides All Common Divisors

The GCD Divides All Common Divisors

Proving the Uniqueness of the Division Algorithm

Proving the Uniqueness of the Division Algorithm

Applying the Division Algorithm

Applying the Division Algorithm

Equivalence Classes and the Derivative

Equivalence Classes and the Derivative

Divisibility Begets Divisibility

Divisibility Begets Divisibility

Misuse of the Well-Ordering Axiom

Misuse of the Well-Ordering Axiom

Proving Sets are Equal

Proving Sets are Equal

Divisible by 0

Divisible by 0

Sets Within Sets

Sets Within Sets

Is it an Equivalence Relation?

Is it an Equivalence Relation?

Intersection of Equivalence Relations

Intersection of Equivalence Relations

What is the Quotient Set

What is the Quotient Set

Finite Groups with Exactly Two Maximal Subgroups are Cyclic

Finite Groups with Exactly Two Maximal Subgroups are Cyclic

The Composition of Surjective Functions is Surjective

The Composition of Surjective Functions is Surjective

Relations, Functions, Injections, Surjections, & Bijections

Relations, Functions, Injections, Surjections, & Bijections

Is it the Empty Set?

Is it the Empty Set?

Unions, Intersections, Complements, and Subsets

Unions, Intersections, Complements, and Subsets

The Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus

The First Fundamental Theorem of Calculus

Groups of Order 2023

Groups of Order 2023

Some Integration Properties

Some Integration Properties

Spread Properties

Spread Properties

Upper and Lower Sums for Functions Sums and Multiples

Upper and Lower Sums for Functions Sums and Multiples

Sup and Inf Omnibus Theorem

Sup and Inf Omnibus Theorem

The Cantor Set and its Indicator Function

The Cantor Set and its Indicator Function

Sequential Characterization of Integrability

Sequential Characterization of Integrability

Thomae's Function

Thomae's Function

Finite Set of Discontinuities Preserves Integrability

Finite Set of Discontinuities Preserves Integrability

Analyzing Particle Dynamics with Derivatives

Analyzing Particle Dynamics with Derivatives

Horizontal Tangent to a Graph

Horizontal Tangent to a Graph

Integration Across a Divided Interval

Integration Across a Divided Interval

Continuity Implies Integrability

Continuity Implies Integrability

The Integrability Criterion

The Integrability Criterion

Darboux Integrability

Darboux Integrability

Upper and Lower Integrals

Upper and Lower Integrals

Lower and Upper Sum Relationships

Lower and Upper Sum Relationships

Lower and Upper Sums

Lower and Upper Sums

Partitions

Partitions

The Generalized Mean Value Theorem

The Generalized Mean Value Theorem

The Mean Value Theorem

The Mean Value Theorem

Rolle's Theorem

Rolle's Theorem

Superquiz 02 Solutions

Superquiz 02 Solutions

The Quaternion Subgroup in M₂(ℂ)

The Quaternion Subgroup in M₂(ℂ)

Fermat's Theorem and Darboux's Theorem

Fermat's Theorem and Darboux's Theorem

Homework 7 Feedback

Homework 7 Feedback

Homework 6 Feedback

Homework 6 Feedback

Differentiability Properties

Differentiability Properties

A Differentiability Consequence and Characterization

A Differentiability Consequence and Characterization

Differentiability Examples

Differentiability Examples

Differentiability Defined

Differentiability Defined

Superquiz 1 Solutions

Superquiz 1 Solutions

YouTube Chapters Display

YouTube Chapters Display

Homework 05 Feedback

Homework 05 Feedback

Continuous Functions on Compact Domains are Uniformly Continuous

Continuous Functions on Compact Domains are Uniformly Continuous

Sequential Characterization of Uniform Continuity

Sequential Characterization of Uniform Continuity

Uniform Continuity

Uniform Continuity

Intermediate Value Property

Intermediate Value Property

Extreme and Intermediate Value Theorems

Extreme and Intermediate Value Theorems

Continuous Functions and Compact Sets

Continuous Functions and Compact Sets

A Topological Characterization of Continuous Functions

A Topological Characterization of Continuous Functions

Homework 04 Feedback

Homework 04 Feedback

A Tough Integral Question

A Tough Integral Question

Some Properties of Continuous Functions

Some Properties of Continuous Functions

A Sequential Characterization of Continuity

A Sequential Characterization of Continuity

Continuity

Continuity

More Properties of Functional Limits

More Properties of Functional Limits

Properties of Functional Limits

Properties of Functional Limits

Functional Limits

Functional Limits

Homework 03 Feedback

Homework 03 Feedback

Compact Sets

Compact Sets

Homework 2 Feedback

Homework 2 Feedback

More On Closed Sets

More On Closed Sets

Closed Sets

Closed Sets

Open Sets

Open Sets

More Convergence Tests

More Convergence Tests

Convergence Tests

Convergence Tests

Infinite Series

Infinite Series

Homework 01 Feedback

Homework 01 Feedback

Cauchy Sequences

Cauchy Sequences

More Limit Properties

More Limit Properties

Bounded and Monotone Sequences

Bounded and Monotone Sequences

Tail and Subsequence Convergence

Tail and Subsequence Convergence

Sequence Convergence

Sequence Convergence

Sequence Basics

Sequence Basics

Cantor's Diagonalization Argument

Cantor's Diagonalization Argument

Countability Omnibus Theorem

Countability Omnibus Theorem

Uncountably Infinite Sets

Uncountably Infinite Sets

Cardinality of the Empty Set

Cardinality of the Empty Set

Countably Infinite Sets

Countably Infinite Sets

Cardinality

Cardinality

Density of the Rationals and Irrationals in the Reals

Density of the Rationals and Irrationals in the Reals

Some Consequences of the Axiom of Completeness

Some Consequences of the Axiom of Completeness

Axiom of Completeness

Axiom of Completeness

Bounds

Bounds

Inequalities

Inequalities

COINs

COINs

Absolute Value

Absolute Value

Set Notation

Set Notation