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.

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

​

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

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

​

A definition of a subspace, with examples.

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.

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.

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

​

Introduction to the method of induction.

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.

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

​

Injectivity, surjectivity, and isomorphism.

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

​

Statement and proof of the rank-nullity theorem.

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.

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

​

Proof that vector spaces of the same dimension are isomorphic.

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.

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

​

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

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.

Endomorphisms of complex vector spaces admit eigenvalues.  Define eigenspaces.

​

Define diagonalizable operators.  State equivalent formulations of diagonalizeability.

Proving some of the equivalent formulations of diagonalizeability.

​

The shape of data.  Geometric interpretations of eigenvectors.

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.

Define norms.  Cauchy-Schwartz inequality, Pythagoras' Theorem, and the Parallelogram Law.

​

Application: Hamming distances.

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

​

The Gram-Schmidt process.

Application of Gram-Schmidt: approximating functions with polynomials.

​

Define the perpendicular to a subset, and properties.

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

​

Recalling dual vector spaces.  The Riesz Representation Theorem. 

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

​

Proof of existence and uniqueness of adjoints.

Define self-adjoint.  Proof that self-adjoint operators admit orthonormal eigenbases.

​

Define normal operator.

Comparing eigenvalues of a normal operator to its adjoint.

​

Statement and proof of the Complex Spectral Theorem.

Approximating operators.  The Frobenius norm and the operator norm.

​

Singular Value Decomposition

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

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

Exterior algebras and the determinant.