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

lambdaofgod/linear_algebra_basics.org