List of topics

Intro

• basic examples - polynomials, $\mathbb{R}^n$ et c
• basis
• independence

Matrices and linear systems

• row space, column space
• criteria for solvability of Ax = 0
• matrix rank
• determinant and linear equations
• trace

Important matrices

• symmetric/hermitian
• antisymmetric
• orthogonal/unitary
• determinants of these matrices

Dot product

• orthonormal basis
• Schwarz inequality
• polarization equality
• isometries - relationship with orthogonal matrices
• positive (semi)definite matrices
• distance from a subspace

Eigenvalues and eigenvectors

• definition
• characteristic polynomial
• relationship with the determinant and trace
• spectra of important matrices

Norms

• basic properties and motivation
• different norms on $\mathbb{R}^n$
• matrix norms
• Banach spaces - preview A vector space with a metric is called Banach space if it is Cauchy complete. The power of this definition is that it trivially includes any finite-dimensional space with a norm, but also many interesting infinite-dimensional spaces.

Decompositions

• eigenvectors and decompositions
• Gram-Schmidt orthogonalization
• QR decomposition
• SVD

Convexity (optional)

• definition and basic properties A convex function on a convex set attains maxima on the boundary
• relationship with differentiability
• Farkas lemma
• intersections of convex sets
  • Radon’s theorem
  • Caratheodory’s theorem
  • Helly’s theorem

Applications

• linear regression - least squares
• principal component analysis
• SVM (requires convexity)
• stochastic matrices
  • Pagerank - definition, algorithm

Advanced theory

• quadratic forms
• Laplacian matrix of a graph
• Jordan normal form
• multilinear algebra - tensors $V \bigotimes W$

Hilbert space basics

Example: $L^2[0, 1]$ space (square integrable functions on an interval), $l_2$ (square summable series) Typically the basic facts about these are introduced in analysis courses but for someone who knows integration it makes sense to include in linear algebra course. Basically this boils down to two facts:

• $∫_[0,1] cos(2 π n x) dx = 1$ and
• $∫_[0,1] cos(2 π n x)cos(2 π m x) dx = δ_mn$

Hilbert spaces are the simplest **infinite dimensional** Banach spaces. Infinite-dimensional spaces are interesting because they might contain proper subspaces that are isomorphic to the whole space.

quantum computing

Quantum computing is based on the Born rule which says for a vector in Hilbert space for which $\|v\|=1$ can be interpreted as a probability distribution. A qubit is a unit vector in $\mathbb{C}^2$ (denoted by $α |0\rangle \rangle + β |1\rangle$). Quantum computing algorithms operate on quantum registers - elements of $\mathbb{C}^{2^n} = \bigotimes_n \mathbb{C}^2$ with unit norm. A quantum gate, linear operator $M$ that maps registers to registers Many interesting properties in quantum computing are elementary facts about finite-dimensional complex vector spaces and unitary operators, for example the no-cloning theorem