## 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 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.