NOTES: STANFORD MATH 113, FALL 2020
Linear algebra and matrix theory
Below are my lecture notes from the remote class, written in realtime over Zoom. View (at your own peril  errors likely abound!) by clicking the dashed arrows. Some supercool stuff is highlighted via yellow text.
References for these lecture notes include Axler's Linear Algebra Done Right, Katznelson's A Terse Introduction to Linear Algebra, math3ma, and various other versions of this course (Akshay Venkatesh, Sheel Ganatra).
Lecture 1

Discussion: What is a proof? When is a proof? Why is a proof?

A definition of a vector space over the real numbers, with examples.
Lecture 2

A definition of a field, finite fields, and vector spaces over fields.

The game of SET reinterpreted as finding lines in a vector space.
Lecture 3

Using the vector space axioms to prove new results. Good proofs and bad proofs.

A definition of a subspace, with examples.
Lecture 4

Conditions for a subset to be a subspace. Examples and nonexamples. The Fibonacci sequence. Sums and direct sums of subspaces.

Proof that span is the smallest containing subspace.
Lecture 5

Continuous functions are a direct sum of even and odd function subspaces.

Definition of basis. Conditions for a list of vectors to be a basis. Steps towards showing that bases exist.
Lecture 6

Steinitz's Lemma. Proof that two bases of a vector space have the same length. Definition of dimension.

Introduction to the method of induction.
Lecture 7

Taking stock: what have we discussed thus far, and how is this data organized in my head?

Definition and examples of linear maps. Bijections. Compositions of linear maps.
Lecture 8

Linear maps as matrices. Linear maps as paths and graphs. Composition as matrix multiplication and path concatenation.

Injectivity, surjectivity, and isomorphism.
Lecture 9

Reframing infectivity and surjectivity in terms of the nullspace and image. Proof via the contrapositive.

Statement and proof of the ranknullity theorem.
Lecture 10

Illustrating the ranknullity theorem with examples and nonexamples.

Applications: there exists a degree d polynomial through d+1 points. Systems of linear equations.
Lecture 11

An example: building intuition and insight as we attempt to figure out a proof.

Proof that vector spaces of the same dimension are isomorphic.
Lecture 12

Define inverses and invertible functions. Examples and nonexamples of invertible functions. Proof that an inverse is unique.

Proof that isomorphisms are precisely invertible linear maps.
Lecture 13

Discussion: how do we get a handle on what a linear map is doing?

Define invariant subspaces, eigenvectors, and eigenvalues, with examples.
Lecture 14

Proof that eigenvectors of distinct eigenvalues are linearly independent. Not all endomorphisms have eigenvalues.

When does something admit a field structure? The fundamental theorem of algebra.
Lecture 15

Endomorphisms of complex vector spaces admit eigenvalues. Define eigenspaces.

Define diagonalizable operators. State equivalent formulations of diagonalizeability.
Lecture 16

Proving some of the equivalent formulations of diagonalizeability.

The shape of data. Geometric interpretations of eigenvectors.
Lecture 17

Angles and dot products in real Euclidean space. How can we generalize this to abstract vector spaces?

Define inner product spaces. Inner products on complex vector spaces & function spaces.
Lecture 18

Define norms. CauchySchwartz inequality, Pythagoras' Theorem, and the Parallelogram Law.

Application: Hamming distances.
Lecture 19

Define orthogonal and orthonormal lists. Illustrate how "norms detect linear independence."

The GramSchmidt process.
Lecture 20

Application of GramSchmidt: approximating functions with polynomials.

Define the perpendicular to a subset, and properties.
Lecture 21

More on perpendiculars. Direct sums and perpendiculars. Orthogonal projection maps.

Recalling dual vector spaces. The Riesz Representation Theorem.
Lecture 22

Matrix transposes and Hermitian conjugates. Define adjoint of linear map.

Proof of existence and uniqueness of adjoints.
Lecture 24

Define selfadjoint. Proof that selfadjoint operators admit orthonormal eigenbases.

Define normal operator.
Lecture 25

Comparing eigenvalues of a normal operator to its adjoint.

Statement and proof of the Complex Spectral Theorem.
Lecture 26

Approximating operators. The Frobenius norm and the operator norm.

Singular Value Decomposition
Lecture 27

Principal Component Analysis, and how it relates to the singular value decomposition: a mathematician's perspective.
Lecture 28

Introduction to tensors. Tensors as multidimensional data; tensors as a mathematical construction.

Exterior algebras and the determinant.