NOTES: STANFORD MATH 113, FALL 2020

Linear algebra and matrix theory

Below are my lecture notes from the remote class, written in real-time over Zoom.  View (at your own peril -- errors likely abound!) by clicking the dashed arrows.  Some super-cool 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 non-examples. 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 rank-nullity theorem.

Lecture 10
  • Illustrating the rank-nullity theorem with examples and non-examples.

  • 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 non-examples 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.  Cauchy-Schwartz inequality, Pythagoras' Theorem, and the Parallelogram Law.

  • Application: Hamming distances.

Lecture 19
  • Define orthogonal and orthonormal lists.  Illustrate how "norms detect linear independence." 

  • The Gram-Schmidt process.

Lecture 20
  • Application of Gram-Schmidt: 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 self-adjoint.  Proof that self-adjoint 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.