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MIT 18.701
Algebra I
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Section 1: Introductory Group Theory
Lecture 1: Matrix and Group Definitions
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Matrix Definition
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Matrix Multiplication and Inverses
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Group Theory Definitions
Lecture 2: Group Examples, Subgroups, Isomorphisms, Homomorphisms
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Group Theory Definitions (cont.)
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Group Examples by Order
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Subgroups
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Geometric Groups
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Isomorphisms
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Group Generation
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Homomorphisms
Lecture 3: Kernel and Image, Cosets, Normal Subgroups, Conjugation, The Correspondence Theorem
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Kernel and Image
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Cosets
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Normal Subgroups
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Conjugation
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Correspondence Theorem
Lecture 4: The Correspondence Theorem, Quotient Groups, The Isomorphism Theorems, Conjugation in
S
n
S_n
S
n
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Proof of the Correspondence Theorem
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Quotient Groups
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The Isomorphism Theorems
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Conjugation and Normality in
S
n
S_n
S
n
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Section 2: Fields and Vector Spaces
Lecture 5: Fields, Vector Spaces, Dimension
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Field Definitions
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Vector Space Definitions
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Basis and Dimension
Lecture 6: Linear Transformations, Bases and Matrices, The Dimension Theorem
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Vector Space Definitions (cont.)
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Choosing a Basis
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Linear Transformation
⟹
\implies
⟹
Matrix (Only if You Choose a Basis!)
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Rank, Nullity, and the Dimension Theorem
Lecture 7: Eigenvectors, Diagonalization, Nilpotence, Jordan Blocks
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Eigenvalues, Eigenvectors, and the Characteristic Polynomial
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Eigenbases and Diagonalization
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Nilpotent Matrices and Jordan Blocks
Lecture 8: Proving Jordan Normal Form, Cayley-Hamilton
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Proving JNF: Assuming Zero is an Eigenvalue
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Proving JNF: Factoring Out a Nilpotent Matrix
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Proving JNF: The Nilpotent Inductive Step
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The Cayley-Hamilton Theorem
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Section 3: Geometric Group Theory
Lecture 9: Inner Products, Orthonormality
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Inner Products in
R
n
\mathbb{R}^n
R
n
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Orthonormal Bases and Matrices
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Orthogonal Groups
Lecture 10: Isometries in
R
2
\mathbb{R}^2
R
2
, Discrete Subgroups of
O
2
(
R
)
O_2(\mathbb{R})
O
2
(
R
)
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Isometries
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Sidenote: Affine Stuffs
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Symmetry Groups
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The Big Four: Isometries in
R
2
\mathbb{R}^2
R
2
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Defining Discreteness
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Classifying Discrete Subgroups of
O
2
(
R
)
O_2(\mathbb{R})
O
2
(
R
)
Lecture 11: Point Groups, Lattices, Crystallographic Restriction, Group Actions
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Point Groups and Lattices
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Classifying Discrete Symmetry Groups: Mathematical Reasoning
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Classifying Discrete Symmetry Groups: The Results
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Group Actions and Cayley's Theorem
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Section 4: Group Actions
Lecture 12: Orbit-Stabilizer, Symmetries of
S
O
3
(
R
)
SO_3(\mathbb{R})
S
O
3
(
R
)
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Orbit and Stabilizer Definitions
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Same Orbit
⟹
\implies
⟹
Conjugate Stabilizers
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Group Actions are Coset Actions
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The Orbit-Stabilizer Theorem
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Classifying Symmetries of
S
O
3
(
R
)
SO_3(\mathbb{R})
S
O
3
(
R
)
Lecture 13: Conjugacy Classes, Centralizer, Center, Class Equation, Extensions, Simple Groups
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Group Actions by Conjugation
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The Class Equation: Classifying Groups of
p
2
p^2
p
2
Order
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Definitions: Group Extensions
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Definitions: We're a Finite Simple Group…
Lecture 14: Icosahedron Symmetries, Jordan-Holder
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Icosahedron Symmetries are Simple
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Composition Series
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The Jordan-Holder Theorem
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The Second Isomorphism Theorem
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The Diamond Lemma
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Proof of The Jordan-Holder Theorem
Lecture 15: Sylow's Theorems
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Stating Sylow's Theorems
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Applications of Sylow's Theorems: Group Classification
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Proving Sylow's Theorems
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Section 5: Bilinear Forms
Lecture 16: Bilinear and Hermitian Forms
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Bilinear Forms: Definitions and Properties
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Bilinear Form
⟹
\implies
⟹
Matrix (Only if You Choose a Basis!)
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Foreshadowing: Quadratic Polynomials are Bilinear Forms
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Hermitian Forms: Doing it all again…
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Transformations that Preserve a Bilinear / Hermitian Form
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Symmetric Bilinear / Hermitian Forms
Lecture 17: Classifying Hermitian Forms, Sylvester's Law of Inertia
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Classifying Hermitian Forms
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Orthogonality and Nondegenerate Subspaces
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Proving Hermitian Classification
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Signatures and Sylvester's Law of Inertia
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Foreshadowing: Signatures are Critical Points…
Lecture 18: Gram-Schmidt, Sylvester's Criterion, The Spectral Theorem
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Inner Product
⟺
\iff
⟺
Orthonormal Basis
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Positive Definite Matrices and Sylvester's Criterion
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Matrix Adjective Review: Hermitian, Unitary, and Normal
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The Spectral Theorem
Lecture 19: Applications of the Spectral Theorem
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Spectral Application #1: Classifying Quadric Surfaces
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Spectral Application #2: Principal Component Analysis
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Spectral Application #3: Singular Value Decomposition
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Linear Groups
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Section 6: Matrix Groups
Lecture 20:
S
U
n
SU_n
S
U
n
and the Quaternions
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Spheres and Balls
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The Special Unitary Group
S
U
2
SU_2
S
U
2
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The Quaternions
H
\mathbb{H}
H
§
Sidenote: Linear Algebra Over
H
\mathbb{H}
H
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Sidenote: A Number Theoretic Application
Lecture 21: Quaternions,
O
4
(
R
)
O_4(\mathbb{R})
O
4
(
R
)
,
S
O
4
(
R
)
SO_4(\mathbb{R})
S
O
4
(
R
)
,
S
O
3
(
R
)
SO_3(\mathbb{R})
S
O
3
(
R
)
, and Conjugacy Classes of
S
U
2
SU_2
S
U
2
§
Quaternion Inner Product and
{
O
4
(
R
)
,
S
O
4
(
R
)
,
S
O
3
(
R
)
}
\{O_4(\mathbb{R}), SO_4(\mathbb{R}), SO_3(\mathbb{R})\}
{
O
4
(
R
)
,
S
O
4
(
R
)
,
S
O
3
(
R
)}
§
Conjugacy Classes of
S
U
2
SU_2
S
U
2
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Geometry of
S
U
2
SU_2
S
U
2
: Latitude and Longitude
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Geometry of
S
U
2
SU_2
S
U
2
: Normal Subgroups and Simplicity
Lecture 22: One-Parameter Subgroups, Lie Algebras, Lie Brackets
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Matrix Exponentiation: One-Parameter
G
L
n
(
C
)
GL_n(\mathbb{C})
G
L
n
(
C
)
Subgroups
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One-Parameter Subgroups of
S
L
n
(
C
)
SL_n(\mathbb{C})
S
L
n
(
C
)
and
O
n
(
R
)
O_n(\mathbb{R})
O
n
(
R
)
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Lie Algebras and Lie Brackets