Bobby W. Lindsey
$ %---- MACROS FOR SETS ----% \newcommand{\znz}[1]{\mathbb{Z} / #1 \mathbb{Z}} \newcommand{\twoheadrightarrowtail}{\mapsto\mathrel{\mspace{-15mu}}\rightarrow} % popular set names \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\I}{\mathbb{I}} % popular vector space notation \newcommand{\V}{\mathbb{V}} \newcommand{\W}{\mathbb{W}} \newcommand{\B}{\mathbb{B}} \newcommand{\D}{\mathbb{D}} %---- MACROS FOR FUNCTIONS ----% % linear algebra \newcommand{\T}{\mathrm{T}} \renewcommand{\ker}{\mathrm{ker}} \newcommand{\range}{\mathrm{range}} \renewcommand{\span}{\mathrm{span}} \newcommand{\rref}{\mathrm{rref}} \renewcommand{\dim}{\mathrm{dim}} \newcommand{\col}{\mathrm{col}} \newcommand{\nullspace}{\mathrm{null}} \newcommand{\row}{\mathrm{row}} \newcommand{\rank}{\mathrm{rank}} \newcommand{\nullity}{\mathrm{nullity}} \renewcommand{\det}{\mathrm{det}} \newcommand{\proj}{\mathrm{proj}} \renewcommand{\H}{\mathrm{H}} \newcommand{\trace}{\mathrm{trace}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\card}{\mathrm{card}} % differential equations \newcommand{\laplace}[1]{\mathcal{L}\{#1\}} \newcommand{\F}{\mathrm{F}} % misc \newcommand{\sign}{\mathrm{sign}} \newcommand{\softmax}{\mathrm{softmax}} \renewcommand{\th}{\mathrm{th}} \newcommand{\adj}{\mathrm{adj}} \newcommand{\hyp}{\mathrm{hyp}} \renewcommand{\max}{\mathrm{max}} \renewcommand{\min}{\mathrm{min}} \newcommand{\where}{\mathrm{\ where\ }} % statistics \newcommand{\cov}{\mathrm{cov}} \newcommand{\var}{\mathrm{var}} \newcommand{\E}{\mathrm{E}} \newcommand{\prob}{\mathrm{prob}} \newcommand{\unif}{\mathrm{unif}} \newcommand{\invNorm}{\mathrm{invNorm}} \newcommand{\invT}{\mathrm{invT}} % real analysis \renewcommand{\sup}{\mathrm{sup}} \renewcommand{\inf}{\mathrm{inf}} %---- MACROS FOR ALIASES AND REFORMATTING ----% % logic \newcommand{\forevery}{\ \forall\ } \newcommand{\OR}{\lor} \newcommand{\AND}{\land} \newcommand{\then}{\implies} % set theory \newcommand{\impropersubset}{\subseteq} \newcommand{\notimpropersubset}{\nsubseteq} \newcommand{\propersubset}{\subset} \newcommand{\notpropersubset}{\not\subset} \newcommand{\union}{\cup} \newcommand{\Union}[2]{\bigcup\limits_{#1}^{#2}} \newcommand{\intersect}{\cap} \newcommand{\Intersect}[2]{\bigcap\limits_{#1}^{#2}} \newcommand{\intersection}[2]{\bigcap\limits_{#1}^{#2}} \newcommand{\Intersection}[2]{\bigcap\limits_{#1}^{#2}} \newcommand{\closure}{\overline} % linear algebra \newcommand{\subspace}{\le} \newcommand{\angles}[1]{\langle #1 \rangle} \newcommand{\identity}{\mathbb{1}} \newcommand{\orthogonal}{\perp} \renewcommand{\parallel}[1]{#1^{||}} % calculus \newcommand{\integral}[2]{\int\limits_{#1}^{#2}} \newcommand{\limit}[1]{\lim\limits_{#1}} \newcommand{\approaches}{\rightarrow} \renewcommand{\to}{\rightarrow} \newcommand{\convergesto}{\rightarrow} % algebra \newcommand{\summation}[2]{\sum\limits_{#1}^{#2}} \newcommand{\product}[2]{\prod\limits_{#1}^{#2}} \newcommand{\by}{\times} % exists commands \newcommand{\notexist}{\nexists\ } \newcommand{\existsatleastone}{\exists\ } \newcommand{\existsonlyone}{\exists!} \newcommand{\existsunique}{\exists!} \let\oldexists\exists \renewcommand{\exists}{\ \oldexists\ } % statistics \newcommand{\distributed}{\sim} \newcommand{\onetoonecorresp}{\sim} \newcommand{\independent}{\perp\!\!\!\perp} \newcommand{\conditionedon}{\ |\ } \newcommand{\given}{\ |\ } \newcommand{\notg}{\ngtr} \newcommand{\yhat}{\hat{y}} \newcommand{\betahat}{\hat{\beta}} \newcommand{\sigmahat}{\hat{\sigma}} \newcommand{\muhat}{\hat{\mu}} \newcommand{\transmatrix}{\mathrm{P}} \renewcommand{\choose}{\binom} % misc \newcommand{\infinity}{\infty} \renewcommand{\bold}{\textbf} \newcommand{\italics}{\textit} $
Math the language of the universe

Preliminary Definitions and Theorems Two points in $E \impropersubset \R$ are rationally equivalent provided their difference belongs to $\Q$; this...

Preliminary Definitions and Theorems A countable collection of sets is ascending provided each set is a subset of the following...

Now that we’re familiar with what measurability is, there are couple other characterizations of measurability that we’ll be going over...

To remedy the predicament we found ourselves in the last post with the outer measure over the real numbers lacking...

Ideally we’d like to have these Lebesgue measurable sets with a Lebesgue measure set function, but it’s actually not possible...

In measure theory, we study ways of generalizing the idea of length, area, and volume. For example, certain functions aren’t...

Stationary distribution As stated in the introduction, a Markov chain is useful for representing stochastic systems because we can infer...

Stochastic models have a variety of applications including Google’s Page Rank algorithm and reinforcement learning algorithms. These models can be...

An eigenvector of a transformation is a non-zero vector where its transformation is a scalar multiple of itself. This associated...

The inner product, denoted <>, on a vector space is a rule that accepts any two vectors and returns a...

Orthogonal vectors and sets Any two vectors are orthogonal if their inner product is equal to 0 and a matrix...

Coordinates and changing bases The coordinates of x relative to a basis B is the vector [x]B = c such...

Properties of a Basis A set of vectors, B, is a basis for a vector space is B is linearly...

The below is a review of linear systems, linear transformations, and the Rank Nullity Theorem. Linear transformations Let V and...

The following is a review of linear dependence, the square matrix theorem, the rank nullity theorem, and the invertible matrix...

In the post, we talk about fields, vector spaces, and subspaces. Fields and vector spaces Suppose F is a nonempty...

For those who haven’t heard of LaTeX, it’s an awesome typesetting language typically used when writing mathematics. As I study...

Markov models are remarkably suited to mimicking the structure of a phenomenon and as a result, I thought it would...

Millions of Americans play the lotto each year as an innocent way to gamble their savings on winning some money...