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9717 lines (7425 loc) · 462 KB
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% ****************************************************************************************
% ************************ ALGEBRA LINEAL ****************************
% ****************************************************************************************
% =======================================================
% ======= HEADER FOR DOCUMENT ============
% =======================================================
% ********* SPECIFIC FOR THIS BOOK ********
\def\ProjectAuthorLink{https://github.com/CompilandoConocimiento}
\def\ProjectNameLink{\ProjectAuthorLink/LibroAlgebraLineal}
% ********* DOCUMENT ITSELF **************
\documentclass[12pt, fleqn]{report} %Type of doc and size of font and left equations
\usepackage[margin=1.2in]{geometry} %Margins and Geometry pacakge
\usepackage{ifthen} %Allow simple programming using if - then
\usepackage[hidelinks]{hyperref} %Allow to create hiperlinks and Fuck Firefox
\usepackage{pdfpages} %Allow us 'import' PDF's
\hypersetup{pageanchor=false} %Solve 'double page 1' warnings in build :v
\setlength{\parindent}{0pt} %Eliminate ugly indentation
\author{Oscar Andrés Rosas} %Who I am
% ********* LANGUAJE *****************
\usepackage[spanish]{babel} %Please allow me to type in spanish
\usepackage[utf8]{inputenc} %Lets use UFT-8
\usepackage[T1]{fontenc} %Allow for better font support
\usepackage{textcmds} %Allow us to use quoutes
\usepackage{changepage} %Allow us to use identate paragraphs
\usepackage{anyfontsize} %All the sizes for fonts wiiiii!
% ********* MATH AND HIS STYLE *********
\usepackage{ntheorem, amsmath, amssymb, amsfonts} %All fucking math, I want all!
\usepackage{mathrsfs, mathtools, empheq} %All fucking math, I want all!
\usepackage{cancel} %Negate symbol
\usepackage{centernot} %Allow me to negate a symbol
\decimalpoint %Use decimal point
% ********* GRAPHICS AND IMAGES *********
\usepackage{graphicx} %Allow to create graphics
\usepackage{float} %For images
\usepackage{wrapfig} %Allow to create images
\graphicspath{ {Graphics/} } %Where are the images :D
% ********* LISTS AND TABLES ***********
\usepackage{listings, listingsutf8} %We will be using code here
\usepackage[inline]{enumitem} %We will need to enumarate
\usepackage{tasks} %Horizontal lists
\usepackage{longtable} %Lets make tables awesome
\usepackage{booktabs} %Lets make tables awesome
\usepackage{tabularx} %Lets make tables awesome
\usepackage{multirow} %Lets make tables awesome
\usepackage{multicol} %Create multicolumns
% ********* REMOVE SOME ERRORS **********
\hbadness=10000 %Ignore \vbox and \hbox warings
\hfuzz=\maxdimen\newdimen\hfuzz %Ignore \vbox and \hbox warings
% ********* HEADERS AND FOOTERS ********
\usepackage{fancyhdr} %Lets make awesome headers/footers
\pagestyle{fancy} %Lets make awesome headers/footers
\setlength{\headheight}{16pt} %Top line
\setlength{\parskip}{0.5em} %Top line
\renewcommand{\footrulewidth}{0.5pt} %Bottom line
\lhead { %Left Header
\hyperlink{chapter.\arabic{chapter}} %Make a link to the current chapter
{\normalsize{\textsc{\nouppercase{\leftmark}}}} %And fot it put the name
}
\rhead { %Right Header
\hyperlink{section.\arabic{chapter}.\arabic{section}} %Make a link to the current chapter
{\footnotesize{\textsc{\nouppercase{\rightmark}}}} %And fot it put the name
}
\rfoot{\textsc{\small{\hyperref[sec:Index]{Ve al Índice}}}} %This will always be a footer
\fancyfoot[L]{ %Algoritm for a changing footer
\ifthenelse{\isodd{\value{page}}} %IF ODD PAGE:
{\href{https://SoyOscarRH.github.io/} %DO THIS:
{\footnotesize %Send the page
{\textsc{Oscar Andrés Rosas}}}} %Send the page
{\href{https://compilandoconocimiento.com} %ELSE DO THIS:
{\footnotesize %Send the author
{\textsc{Compilando Conocimiento}}}} %Send the author
}
% =======================================================
% =================== COMMANDS =====================
% =======================================================
% =========================================
% ======= NEW ENVIRONMENTS ============
% =========================================
\newenvironment{Indentation}[1][0.75em] %Use: \begin{Inde...}[Num]...\end{Inde...}
{\begin{adjustwidth}{#1}{}} %If you dont put nothing i will use 0.75 em
{\end{adjustwidth}} %This indentate a paragraph
\newenvironment{SmallIndentation}[1][0.75em] %Use: The same that we upper one, just
{\begin{adjustwidth}{#1}{}\begin{footnotesize}} %footnotesize size of letter by default
{\end{footnotesize}\end{adjustwidth}} %that's it
\def \Eq {equation} %Stupid Visual studio error
\newenvironment{MultiLineEquation}[1] %Use: To create MultiLine equations
{\begin{\Eq}\begin{alignedat}{#1}} %Use: \begin{Multi..}{Num. de Columnas}
{\end{alignedat}\end{\Eq}} %And.. that's it!
\newenvironment{MultiLineEquation*}[1] %Use: To create MultiLine equations
{\begin{\Eq*}\begin{alignedat}{#1}} %Use: \begin{Multi..}{Num. de Columnas}
{\end{alignedat}\end{\Eq*}} %And.. that's it!
\newenvironment{largeEq} {\begingroup \large}{\endgroup} %Make eq bigger
\newenvironment{LargeEq} {\begingroup \Large}{\endgroup} %Make eq bigger
\newenvironment{HugeEq} {\begingroup \Huge}{\endgroup} %Make eq bigger!
% =========================================
% == GENERAL TEXT & SYMBOLS ENVIRONMENTS ==
% =========================================
% ===== TEXT ======================
\newcommand \Quote {\qq} %Use: \Quote to use quotes
\newcommand \Over {\overline} %Use: \Bar to use just for short
\newcommand \ForceNewLine {$\Space$\\} %Use it in theorems for example
\newcommand \ForceColumnBreak {\vfill\null\columnbreak} %Use only in multicols
% ===== SPACES ====================
\DeclareMathOperator \Space {\quad} %Use: \Space for a cool mega space
\DeclareMathOperator \MegaSpace {\quad \quad} %Use: \MegaSpace for a cool mega mega space
\DeclareMathOperator \MiniSpace {\;} %Use: \Space for a cool mini space
% ===== MATH TEXT =================
\newcommand \Such {\MiniSpace | \MiniSpace} %Use: \Such like in sets
\newcommand \Also {\MiniSpace \text{y} \MiniSpace} %Use: \Also so it's look cool
\newcommand \Remember[1] {\Space\text{\scriptsize{#1}}} %Use: \Remember so it's look cool
% ===== THEOREMS: IN SPANISH :0 ===
\newtheorem{Theorem} {Teorema}[section] %Use: \begin{Theorem}[Name]\label{Nombre}...
\newtheorem{Corollary} {Colorario}[Theorem] %Use: \begin{Corollary}[Name]\label{Nombre}...
\newtheorem{Lemma}[Theorem] {Lemma} %Use: \begin{Lemma}[Name]\label{Nombre}...
\newtheorem{Definition} {Definición}[section] %Use: \begin{Definition}[Name]\label{Nombre}...
\theoremstyle{break} %THEOREMS START 1 SPACE AFTER Fuck!
% ===== LOGIC =====================
\newcommand \lIff {\leftrightarrow} %Use: \lIff for logic iff
\newcommand \lEqual {\MiniSpace \Leftrightarrow \MiniSpace} %Use: \lEqual for a logic double arrow
\newcommand \lInfire {\MiniSpace \Rightarrow \MiniSpace} %Use: \lInfire for a logic infire
\newcommand \lLongTo {\longrightarrow} %Use: \lLongTo for a long arrow
\newcommand \lAnd {\land} %Use: \lAnd ^
\newcommand \lOr {\lor} %Use: \lOr or symbol
\newcommand \lNot {\neg} %Use: \lNot for negation
% ===== FAMOUS SETS ===============
\DeclareMathOperator \Naturals {\mathbb{N}} %Use: \Naturals por Notation
\DeclareMathOperator \Primes {\mathbb{P}} %Use: \Primes por Notation
\DeclareMathOperator \Integers {\mathbb{Z}} %Use: \Integers por Notation
\DeclareMathOperator \Racionals {\mathbb{Q}} %Use: \Racionals por Notation
\DeclareMathOperator \Reals {\mathbb{R}} %Use: \Reals por Notation
\DeclareMathOperator \Complexs {\mathbb{C}} %Use: \Complex por Notation
\DeclareMathOperator \GenericField {\mathbb{F}} %Use: \GenericField por Notation
\DeclareMathOperator \VectorSet {\mathbb{V}} %Use: \VectorSet por Notation
\DeclareMathOperator \SubVectorSet {\mathbb{W}} %Use: \SubVectorSet por Notation
\DeclareMathOperator \Polynomials {\mathbb{P}} %Use: \Polynomials por Notation
\DeclareMathOperator \VectorSpace {\VectorSet_{\GenericField}} %Use: \VectorSpace por Notation
\DeclareMathOperator \LinealTransformation {\mathcal{T}} %Use: \LinealTransformation for a cool T
\DeclareMathOperator \LinTrans {\mathcal{T}} %Use: \LinTrans for a cool T
\DeclareMathOperator \Laplace {\mathcal{L}} %Use: \LinTrans for a cool T
% ===== CONTAINERS ===============
\newcommand{\Set}[1] {\left\{ \; #1 \; \right\}} %Use: \Set {Info} for INTELLIGENT space
\newcommand{\bigSet}[1] {\big\{ \; #1 \; \big\}} %Use: \bigSet {Info} for space
\newcommand{\BigSet}[1] {\Big\{ \; #1 \; \Big\}} %Use: \BigSet {Info} for space
\newcommand{\biggSet}[1] {\bigg\{ \; #1 \; \bigg\}} %Use: \biggSet {Info} for space
\newcommand{\BiggSet}[1] {\Bigg\{ \; #1 \; \Bigg\}} %Use: \BiggSet {Info} for space
\newcommand{\Wrap}[1] {\left( #1 \right)} %Use: \Wrap {Info} for INTELLIGENT space
\newcommand{\bigWrap}[1] {\big( \; #1 \; \big)} %Use: \bigBrackets {Info} for space
\newcommand{\BigWrap}[1] {\Big( \; #1 \; \Big)} %Use: \BigBrackets {Info} for space
\newcommand{\biggWrap}[1] {\bigg( \; #1 \; \bigg)} %Use: \biggBrackets {Info} for space
\newcommand{\BiggWrap}[1] {\Bigg( \; #1 \; \Bigg)} %Use: \BiggBrackets {Info} for space
\newcommand{\Brackets}[1] {\left[ #1 \right]} %Use: \Brackets {Info} for INTELLIGENT space
\newcommand{\bigBrackets}[1] {\big[ \; #1 \; \big]} %Use: \bigBrackets {Info} for space
\newcommand{\BigBrackets}[1] {\Big[ \; #1 \; \Big]} %Use: \BigBrackets {Info} for space
\newcommand{\biggBrackets}[1] {\bigg[ \; #1 \; \bigg]} %Use: \biggBrackets {Info} for space
\newcommand{\BiggBrackets}[1] {\Bigg[ \; #1 \; \Bigg]} %Use: \BiggBrackets {Info} for space
\newcommand{\Generate}[1] {\left\langle #1 \right\rangle} %Use: \Generate {Info} <>
\newcommand{\Floor}[1] {\left \lfloor #1 \right \rfloor} %Use: \Floor {Info} for floor
\newcommand{\Ceil}[1] {\left \lceil #1 \right \rceil } %Use: \Ceil {Info} for ceil
% ===== BETTERS MATH COMMANDS =====
\newcommand{\pfrac}[2] {\Wrap{\dfrac{#1}{#2}}} %Use: Put fractions in parentesis
\newcommand{\Sum} {\displaystyle \sum} %Use: Sum to big sum
\newcommand{\Int} {\displaystyle \int} %Use: Sum to big integral
% =========================================
% ==== LINEAL ALGEBRA & VECTORS ======
% =========================================
% ===== UNIT VECTORS ================
\newcommand{\hati} {\hat{\imath}} %Use: \hati for unit vector
\newcommand{\hatj} {\hat{\jmath}} %Use: \hatj for unit vector
\newcommand{\hatk} {\hat{k}} %Use: \hatk for unit vector
% ===== MAGNITUDE ===================
\newcommand{\abs}[1] {\left\lvert #1 \right\lvert} %Use: \abs{expression} for |x|
\newcommand{\Abs}[1] {\left\lVert #1 \right\lVert} %Use: \Abs{expression} for ||x||
\newcommand{\Mag}[1] {\left| #1 \right|} %Use: \Mag {Info}
\newcommand{\bVec}[1] {\mathbf{#1}} %Use for bold type of vector
\newcommand{\lVec}[1] {\overrightarrow{#1}} %Use for a long arrow over a vector
\newcommand{\uVec}[1] {\mathbf{\hat{#1}}} %Use: Unitary Vector Example: $\uVec{i}
% ===== FN LINEAL TRANSFORMATION ====
\newcommand{\FnLinTrans}[1]{\mathcal{T}\Wrap{#1}} %Use: \FnLinTrans for a cool T
\newcommand{\VecLinTrans}[1]{\mathcal{T}\pVector{#1}} %Use: \LinTrans for a cool T
\newcommand{\FnLinealTransformation}[1]{\mathcal{T}\Wrap{#1}} %Use: \FnLinealTransformation
% ===== ALL FOR DOT PRODUCT =========
\makeatletter %WTF! IS THIS
\newcommand*\dotP{\mathpalette\dotP@{.5}} %Use: \dotP for dot product
\newcommand*\dotP@[2] {\mathbin { %WTF! IS THIS
\vcenter{\hbox{\scalebox{#2}{$\m@th#1\bullet$}}}} %WTF! IS THIS
} %WTF! IS THIS
\makeatother %WTF! IS THIS
\newcommand{\adotP}[2] {\left< #1, #2 \right> } %Use for <x, y>
\newcommand{\wdotP}[2] {\Wrap{ #1, #2 } } %Use for (x, y)
\newcommand{\cdotP}[2] {\Wrap{ #1 \dotP #2 } } %Use for (x * y)
% === WRAPPERS FOR COLUMN VECTOR ===
\newcommand{\pVector}[1] %Use: \pVector {Matrix Notation} use parentesis
{ \ensuremath{\begin{pmatrix}#1\end{pmatrix}} } %Example: \pVector{a\\b\\c} or \pVector{a&b&c}
\newcommand{\lVector}[1] %Use: \lVector {Matrix Notation} use a abs
{ \ensuremath{\begin{vmatrix}#1\end{vmatrix}} } %Example: \lVector{a\\b\\c} or \lVector{a&b&c}
\newcommand{\bVector}[1] %Use: \bVector {Matrix Notation} use a brackets
{ \ensuremath{\begin{bmatrix}#1\end{bmatrix}} } %Example: \bVector{a\\b\\c} or \bVector{a&b&c}
\newcommand{\Vector}[1] %Use: \Vector {Matrix Notation} no parentesis
{ \ensuremath{\begin{matrix}#1\end{matrix}} } %Example: \Vector{a\\b\\c} or \Vector{a&b&c}
% === MAKE MATRIX BETTER =========
\makeatletter %Example: \begin{matrix}[cc|c]
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c] { %WTF! IS THIS
\hskip -\arraycolsep %WTF! IS THIS
\let\@ifnextchar\new@ifnextchar %WTF! IS THIS
\array{#1} %WTF! IS THIS
} %WTF! IS THIS
\makeatother %WTF! IS THIS
% =========================================
% ======= FAMOUS FUNCTIONS ============
% =========================================
% == TRIGONOMETRIC FUNCTIONS ====
\newcommand{\Cos}[1] {\cos\Wrap{#1}} %Simple wrappers
\newcommand{\Sin}[1] {\sin\Wrap{#1}} %Simple wrappers
\newcommand{\Tan}[1] {tan\Wrap{#1}} %Simple wrappers
\newcommand{\Sec}[1] {sec\Wrap{#1}} %Simple wrappers
\newcommand{\Csc}[1] {csc\Wrap{#1}} %Simple wrappers
\newcommand{\Cot}[1] {cot\Wrap{#1}} %Simple wrappers
% === COMPLEX ANALYSIS TRIG ======
\newcommand \Cis[1] {\Cos{#1} + i \Sin{#1}} %Use: \Cis for cos(x) + i sin(x)
\newcommand \pCis[1] {\Wrap{\Cis{#1}}} %Use: \pCis for the same with parantesis
\newcommand \bCis[1] {\Brackets{\Cis{#1}}} %Use: \bCis for the same with Brackets
% =========================================
% =========== CALCULUS ============
% =========================================
% ====== TRANSFORMS =============
\newcommand{\FourierT}[1] {\mathscr{F} \left\{ #1 \right\} } %Use: \FourierT {Funtion}
\newcommand{\InvFourierT}[1]{\mathscr{F}^{-1}\left\{#1\right\}} %Use: \InvFourierT {Funtion}
% ====== DERIVATIVES ============
\newcommand \MiniDerivate[1][x] {\dfrac{d}{d #1}} %Use: \MiniDerivate[var] for simple use [var]
\newcommand \Derivate[2] {\dfrac{d \; #1}{d #2}} %Use: \Derivate [f(x)][x]
\newcommand \MiniUpperDerivate[2] {\dfrac{d^{#2}}{d#1^{#2}}} %Mini Derivate High Orden Derivate -- [x][pow]
\newcommand \UpperDerivate[3] {\dfrac{d^{#3} \; #1}{d#2^{#3}}} %Complete High Orden Derivate -- [f(x)][x][pow]
\newcommand \MiniPartial[1][x] {\dfrac{\partial}{\partial #1}} %Use: \MiniDerivate for simple use [var]
\newcommand \Partial[2] {\dfrac{\partial \; #1}{\partial #2}} %Complete Partial Derivate -- [f(x)][x]
\newcommand \MiniUpperPartial[2] %Mini Derivate High Orden Derivate -- [x][pow]
{\dfrac{\partial^{#2}}{\partial #1^{#2}}} %Mini Derivate High Orden Derivate
\newcommand \UpperPartial[3] %Complete High Orden Derivate -- [f(x)][x][pow]
{\dfrac{\partial^{#3} \; #1}{\partial#2^{#3}}} %Use: \UpperDerivate for simple use
\DeclareMathOperator \Evaluate {\Big|} %Use: \Evaluate por Notation
% ====== INTEGRALS ============
\newcommand{\inftyInt} {\int_{-\infty}^{\infty}} %Use: \inftyInt for simple integrants
% =======================================================
% =========== COLOR: MATERIAL DESIGN ===========
% =======================================================
% ===== COLORS ==================
\definecolor{RedMD}{HTML}{F44336} %Use: Color :D
\definecolor{Red100MD}{HTML}{FFCDD2} %Use: Color :D
\definecolor{Red200MD}{HTML}{EF9A9A} %Use: Color :D
\definecolor{Red300MD}{HTML}{E57373} %Use: Color :D
\definecolor{Red700MD}{HTML}{D32F2F} %Use: Color :D
\definecolor{PurpleMD}{HTML}{9C27B0} %Use: Color :D
\definecolor{Purple100MD}{HTML}{E1BEE7} %Use: Color :D
\definecolor{Purple200MD}{HTML}{EF9A9A} %Use: Color :D
\definecolor{Purple300MD}{HTML}{BA68C8} %Use: Color :D
\definecolor{Purple700MD}{HTML}{7B1FA2} %Use: Color :D
\definecolor{IndigoMD}{HTML}{3F51B5} %Use: Color :D
\definecolor{Indigo100MD}{HTML}{C5CAE9} %Use: Color :D
\definecolor{Indigo200MD}{HTML}{9FA8DA} %Use: Color :D
\definecolor{Indigo300MD}{HTML}{7986CB} %Use: Color :D
\definecolor{Indigo700MD}{HTML}{303F9F} %Use: Color :D
\definecolor{BlueMD}{HTML}{2196F3} %Use: Color :D
\definecolor{Blue100MD}{HTML}{BBDEFB} %Use: Color :D
\definecolor{Blue200MD}{HTML}{90CAF9} %Use: Color :D
\definecolor{Blue300MD}{HTML}{64B5F6} %Use: Color :D
\definecolor{Blue700MD}{HTML}{1976D2} %Use: Color :D
\definecolor{Blue900MD}{HTML}{0D47A1} %Use: Color :D
\definecolor{CyanMD}{HTML}{00BCD4} %Use: Color :D
\definecolor{Cyan100MD}{HTML}{B2EBF2} %Use: Color :D
\definecolor{Cyan200MD}{HTML}{80DEEA} %Use: Color :D
\definecolor{Cyan300MD}{HTML}{4DD0E1} %Use: Color :D
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\definecolor{Cyan900MD}{HTML}{006064} %Use: Color :D
\definecolor{TealMD}{HTML}{009688} %Use: Color :D
\definecolor{Teal100MD}{HTML}{B2DFDB} %Use: Color :D
\definecolor{Teal200MD}{HTML}{80CBC4} %Use: Color :D
\definecolor{Teal300MD}{HTML}{4DB6AC} %Use: Color :D
\definecolor{Teal700MD}{HTML}{00796B} %Use: Color :D
\definecolor{Teal900MD}{HTML}{004D40} %Use: Color :D
\definecolor{GreenMD}{HTML}{4CAF50} %Use: Color :D
\definecolor{Green100MD}{HTML}{C8E6C9} %Use: Color :D
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\definecolor{Green300MD}{HTML}{81C784} %Use: Color :D
\definecolor{Green700MD}{HTML}{388E3C} %Use: Color :D
\definecolor{Green900MD}{HTML}{1B5E20} %Use: Color :D
\definecolor{AmberMD}{HTML}{FFC107} %Use: Color :D
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\definecolor{Amber200MD}{HTML}{FFE082} %Use: Color :D
\definecolor{Amber300MD}{HTML}{FFD54F} %Use: Color :D
\definecolor{Amber700MD}{HTML}{FFA000} %Use: Color :D
\definecolor{Amber900MD}{HTML}{FF6F00} %Use: Color :D
\definecolor{OrangeMD}{HTML}{ff9800} %Use: Color :D
\definecolor{Orange100MD}{HTML}{ffe0b2} %Use: Color :D
\definecolor{Orange200MD}{HTML}{ffcc80} %Use: Color :D
\definecolor{Orange300MD}{HTML}{ffb74d} %Use: Color :D
\definecolor{Orange700MD}{HTML}{fb8c00} %Use: Color :D
\definecolor{Orange900MD}{HTML}{ef6c00} %Use: Color :D
\definecolor{BlueGreyMD}{HTML}{607D8B} %Use: Color :D
\definecolor{BlueGrey100MD}{HTML}{CFD8DC} %Use: Color :D
\definecolor{BlueGrey200MD}{HTML}{B0BEC5} %Use: Color :D
\definecolor{BlueGrey300MD}{HTML}{90A4AE} %Use: Color :D
\definecolor{BlueGrey700MD}{HTML}{455A64} %Use: Color :D
\definecolor{BlueGrey900MD}{HTML}{263238} %Use: Color :D
\definecolor{DeepPurpleMD}{HTML}{673AB7} %Use: Color :D
% ===== ENVIRONMENT ==============
\newcommand{\Color}[2]{\textcolor{#1}{#2}} %Simple color environment
\newenvironment{ColorText}[1] %Use: \begin{ColorText}
{ \leavevmode\color{#1}\ignorespaces } %That's is!
% =======================================================
% =========== CODE EDITING ===========
% =======================================================
% ===== CODE EDITOR =============
\lstdefinestyle{CompilandoStyle} { %This is Code Style
backgroundcolor = \color{BlueGrey900MD}, %Background Color
basicstyle = \tiny\color{white}, %Style of text
commentstyle = \color{BlueGrey200MD}, %Comment style
stringstyle = \color{Green300MD}, %String style
keywordstyle = \color{Blue300MD}, %keywords style
numberstyle = \tiny\color{TealMD}, %Size of a number
frame = shadowbox, %Adds a frame around the code
breakatwhitespace = true, %Style
breaklines = true, %Style
showstringspaces = false, %Hate those spaces
breaklines = true, %Style
keepspaces = true, %Style
numbers = left, %Style
numbersep = 10pt, %Style
xleftmargin = \parindent, %Style
tabsize = 4, %Style
inputencoding = utf8/latin1 %Allow me to use special chars
}
% ===== CODE EDITOR =============
\lstdefinestyle{CompilandoStylePurity} { %This is Code Style
backgroundcolor = \color{white}, %Background Color
basicstyle = \tiny\color{BlueGrey900MD}, %Style of text
commentstyle = \color{Green300MD}, %Comment style
stringstyle = \color{Teal700MD}, %String style
keywordstyle = \color{Blue700MD}, %keywords style
numberstyle = \tiny\color{TealMD}, %Size of a number
frame = none, %Adds a frame around the code
breakatwhitespace = true, %Style
breaklines = true, %Style
showstringspaces = false, %Hate those spaces
breaklines = true, %Style
keepspaces = true, %Style
numbers = left, %Style
numbersep = 11pt, %Style
xleftmargin = \parindent, %Style
tabsize = 4, %Style
inputencoding = utf8/latin1 %Allow me to use special chars
}
\lstset{style = CompilandoStyle} %Use this style
% =======================================================
% =========== VARIABLES ===========
% =======================================================
% ===== SOLVE LINEAL SYSTEM =============
\newcommand \ColorMatrixA {Teal700MD} %Color definition
\newcommand \ColorVecX {Green700MD} %Color definition
\newcommand \ColorVecB {Red700MD} %Color definition
\newcommand \ColorMatrixL {Amber700MD} %Color definition
\newcommand \ColorMatrixU {Indigo700MD} %Color definition
\newcommand \ColorVecSemiX {Green900MD} %Color definition
\newcommand \ColorVecSemiB {Red900MD} %Color definition
\newcommand \ColorVecY {Purple700MD} %Color definition
\newcommand \ColorMatrixP {Cyan900MD} %Color definition
\newcommand \ColorMatrixQ {Orange700MD} %Color definition
\newcommand \ColorMatrixR {BlueGrey700MD} %Color definition
\newcommand \MatrixA {\Color{\ColorMatrixA}{A}} %Use: Ax = b
\newcommand \VecX {\Color{\ColorVecX}{\vec x}} %Use: Ax = b
\newcommand \VecXT {\Color{\ColorVecX}{\VecX^{\;t}}} %Use: Ax = b
\newcommand \VecB {\Color{\ColorVecB}{\vec b}} %Use: Ax = b
\newcommand \MatrixL {\Color{\ColorMatrixL}{L}} %Use: A = LU
\newcommand \MatrixLT {\Color{\ColorMatrixL}{L^{\;t}}} %Use: A = LU
\newcommand \MatrixU {\Color{\ColorMatrixU}{U}} %Use: A = LU
\newcommand \VecSemiX {\Color{\ColorVecSemiX}{\tilde x}} %Use: Ax = b
\newcommand \VecSemiB {\Color{\ColorVecSemiB}{\tilde b}} %Use: Ax = b
\newcommand \VecY {\Color{\ColorVecY}{\vec y}} %Use: Ax = b
\newcommand \MatrixP {\Color{\ColorMatrixP}{P}} %Use: PA = LU
\newcommand \MatrixQ {\Color{\ColorMatrixQ}{Q}} %Use: QR = b
\newcommand \MatrixR {\Color{\ColorMatrixR}{R}} %Use: QR = b
% =====================================================
% ============ COVER PAGE ================
% =====================================================
\begin{document}
\begin{titlepage}
% ============ TITLE PAGE STYLE ================
\definecolor{TitlePageColor}{cmyk}{1,.60,0,.40} %Simple colors
\definecolor{ColorSubtext}{cmyk}{1,.50,0,.10} %Simple colors
\newgeometry{left=0.25\textwidth} %Defines an Offset
\pagecolor{TitlePageColor} %Make it this Color to page
\color{white} %General things should be white
% ===== MAKE SOME SPACE =========
\vspace %Give some space
\baselineskip %But we need this to up command
% ============ NAME OF THE PROJECT ============
\makebox[0pt][l]{\rule{1.3\textwidth}{3pt}} %Make a cool line
\href{https://compilandoconocimiento.com} %Link to project
{\textbf{\textsc{\Huge Compilando Conocimiento}}}\\[2.7cm] %Name of project
% ============ NAME OF THE BOOK ===============
\href{\ProjectNameLink} %Link to Author
{\fontsize{45}{58}\selectfont \textbf{Álgebra Lineal y \\ Análisis Númerico}}\\[0.5cm] %Name of the book
\textcolor{ColorSubtext}{\textsc{\Huge Matemáticas}} %Name of the general theme
\vfill %Fill the space
% ============ NAME OF THE AUTHOR =============
\href{\ProjectAuthorLink} %Link to Author
{\LARGE \textsf{Oscar Andrés Rosas Hernandez}} %Author
% ===== MAKE SOME SPACE =========
\vspace %Give some space
\baselineskip %But we need this to up command
{\large \textsf{Octubre 2018}} %Date
\end{titlepage}
% =====================================================
% ========== RESTORE TO DOCUMENT ============
% =====================================================
\restoregeometry %Restores the geometry
\nopagecolor %Use to restore the color to white
% =====================================================
% ======== INDICE =========
% =====================================================
\tableofcontents{}
\label{sec:Index}
\clearpage
% ////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
% /////////////////////// ¿QUE ES LO QUE ESTOY LEYENDO? //////////////////////////////////////////
% ////////////////////////////////////////////////////////////////////////////////////////////////////////////////////
\section{¿Qué es lo que estoy leyendo?}
Hola... ¡Hey! Seguramente te estarás preguntando
¿Qué demonios estoy leyendo?
Bueno, este pequeño texto intenta darle solución a esa pregunta, la respuesta mas inmediata es
que este texto (o compilado como nos gusta decirle) es una recopilación de teoremas, ideas
y conceptos importantes que aprendí a lo largo del tiempo sobre este tema.
De manera regular estarémos actualizando estos textos con todo aquello nuevo que aprenda intentando
profundizar en todos estos temas y cerrar posibles dudas en estas páginas, así que siempre mantente
alerta de tener la última versión, esta siempre esta en \href{http://www.CompilandoConocimiento.com}
{\underline{CompilandoConocimiento.com}}
Este Compilado intenta ser lo más estricto posible, aunque somos humanos y es posible (e incluso probable) que
cometamos pequeños errores de vez en cuando.
Estos textos están creados como una base con la que tu puedas leer rápidamente todo lo que hemos aprendido
a lo largo del tiempo, aprender los conceptos más importantes y que usándo esto tu puedas profundizar
más en la maravilla que es aprender más sobre este maravilloso mundo.
Este texto esta publicado bajo la GPL, por lo tanto es software libre y tu tienes el control total sobre
el, puedes descargar este texto, puedes ver su código fuente, puedes modificarlo y puedes distribuir este
texto y sus versiones modificadas, puedes acceder a todo lo que necesitas
\href{http://www.github.com/CompilandoConocimiento/LibroAlgebraLineal}
{\underline{en el Repositorio del Libro de Algebra Lineal}}.
Cualquier pregunta, comentario o si quieres contactar con nosotros no dudes en escribir al email del proyecto:
CompilandoConocimiento@gmail.com
Espero que tomes estas páginas como un regalo, creado por seres imperfectos pero con muchos ánimos de hacer
del mundo un lugar mejor, ahora si, abróchate los cinturones que esto acaba de empezar.
\begin{flushright}
Compilar es Compartir
\end{flushright}
% //////////////////////////////////////////////////////////////////////////////////////////////////////////
% ///////////////////// INTRODUCCION DE LAS MATRICES ///////////////////////////////
% //////////////////////////////////////////////////////////////////////////////////////////////////////////
\part{Introducción A Matrices}
\clearpage
% ===============================================================================
% =================== ENTENDAMOS A LAS MATRICES ======================
% ===============================================================================
\chapter{Conozcamos las Matrices}
% ==============================================================
% ================= DEFINICION ==================
% ==============================================================
\clearpage
\section{Definición}
Siendo formales una Matriz es un arreglo rectangular de $m \times n$ elementos
(donde $m,n \in \Naturals$), es decir es un objecto matemático de $m$ filas y
de $n$ columnas. \textbf{Repito es un objeto de $m$ filas y de $n$ columnas}.
Las entradas de matrices pueden ser números u objetos más complicados.
\begin{equation*}
A =
\begin{bmatrix}[ccc]
a _{1, 1} & \cdots & a_{1,n} \\
\cdots & & \cdots \\
a _{m, 1} & \cdots & a_{m,n} \\
\end{bmatrix}
\end{equation*}
Sea $\GenericField$ un conjunto (ya se que en mate, tecnicamente todo el un conjunto),
entonces decimos que $M_{m \times n}(\GenericField)$ denota al conjunto de todas las
matrices de tamaños $m \times n$ cuyas entradas pertenecen a $\GenericField$.
% ====================================
% ===== DEFINICION FORMAL ======
% ====================================
\vspace{2em}
\subsection*{Definición más Formal}
Una matriz de tamaño $m \times n$ con elementos en el conjunto $\GenericField$ se puede
definir también como una función que toma un par ordenado (las coordenadas) y regresa
un elemento de $\GenericField$:
\begin{equation*}
\Set{1, \dots, m} \times \Set{1, \dots , n}
\Space \lLongTo \Space
\GenericField
\end{equation*}
% ====================================
% ===== SIMBOLOGIA HERMOSA ======
% ====================================
\clearpage
\subsection{Notación de Matrices mediante Función}
La notación más rara y al mismo tiempo más increíble es:
\begin{equation*}
A
= \BigBrackets{ f(i,j) }_{i, j = 1}^{m, n}
=
\begin{bmatrix}[ccc]
f(1,1) & \cdots & f(1,n) \\
\cdots & & \cdots \\
f(m, 1) & \cdots & f(m,n) \\
\end{bmatrix}
\end{equation*}
Esta notación nos dice que $A$ es una matriz de tamaño $m \times n$ tal
que su entrada ubicada en la fila número $i$ y en la columna $j$ es igual
a la función:
$f: \{1, \dots, m\} \times \{1, \dots, n\} \to \GenericField$
Aquí $f(i, j)$ es una función de dos argumentos.
% ==============================================================
% ================= SIMBOLOGIA =============
% ==============================================================
\vspace{1em}
\section{Simbología y Notación}
Solemos denotar con letras mayúsculas a las matrices y con letras miniscúlas
a cada uno de los elementos.
Para hablar de un elemento en específico usamos $a_{i,j}$ donde $i$ es el
número de fila y $j$ es el número de columnas, o bien podemos escribir $[A]_{i,j}$
\textbf{Recuerda que soy computólogo, así que mis índices pueden empiezar en 0}
% =============================
% ======== EJEMPLO ======
% =============================
\subsubsection*{Ejemplo}
\begin{SmallIndentation}[1em]
Por ejemplo, una matriz sería:
\begin{equation*}
A =
\begin{bmatrix}[ccc]
a & b & c \\
d & e & f \\
\end{bmatrix}
\end{equation*}
y $a_{1,3}$ ó $[A]_{1, 3}$ es el elemento $c$.
\end{SmallIndentation}
% ==============================================================
% ================ DELTA DE KRONECKER =============
% ==============================================================
\vspace{1em}
\section{Delta de Kronecker}
Esta es una función demasiado sencilla $\delta(i,j): \Naturals^2 \to \{0,1\}$
pero muy importante a lo largo de Álgebra Lineal, podemos definirla como:
\begin{equation*}
\delta(i,j) =
\begin{cases}
1 \Space \text{ si } i = j \\
0 \Space \text{ si } i \neq j
\end{cases}
\end{equation*}
% ==============================================================
% ================ CLASIFICACION DE MATRICES =============
% ==============================================================
\clearpage
\section{Clasificación y Matrices Famosas}
% ===================================
% ======= MATRICES CUADRADAS =====
% ===================================
\subsection{Matrices Cuadradas}
Son aquellas matrices de $m \times n$ donde $m = n$.
Solemos decir que el orden de estas matrices es $n$.
Por ejemplo:
\begin{equation*}
A_{n \times n} =
\begin{bmatrix}[ccc]
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{bmatrix}
\end{equation*}
Solemos decir que cualquier matriz que no sea cuadrada es
rectangular, es decir son aquellas matrices de $m \times n$
si es que $m \neq n$.
Es importante hablar de las matrices cuadradas porque hay muchas
características que solo funcionan si tu matriz es cuadrada.
% ===================================
% ======= MATRICES IDENTIDAD ====
% ===================================
\clearpage
\subsection{Matriz Identidad: $I_n$}
Son todas las matrices cuadradas donde cada elemento cumple que:
\begin{equation*}
[I]_{i, j} = \delta(i, j)
\end{equation*}
O más formalmente podemos definir a la Matriz identidad de órden $n$ como:
\begin{equation*}
\BigBrackets{\delta(i,j)}_{i, j = 1}^{n, n}
\end{equation*}
Se ve algo así:
\begin{equation*}
I_n =
\begin{bmatrix}[cccc]
1 & 0 & \dots & 0 \\
0 & 1 & \dots & 0 \\
\vdots \\
0 & 0 & \dots & 1 \\
\end{bmatrix}
\end{equation*}
% ===================================
% ======= MATRICES CERO ====
% ===================================
\vspace{2em}
\subsection{Matriz Cero: $0_{m \times n}$}
Son todas aquellas matrices $m \times n$ que cumplen que para cada elemento:
\begin{equation*}
[0]_{i,j} = 0
\end{equation*}
O más formalmente podemos definir a la Matriz de Ceros de órden $n$ como:
\begin{equation*}
\BigBrackets{0}_{i, j = 1}^{n, n}
\end{equation*}
Se ven algo así:
\begin{equation*}
0_{m \times n} =
\begin{bmatrix}[cccc]
0 & 0 & \dots & 0 \\
0 & 0 & \dots & 0 \\
\vdots \\
0 & 0 & \dots & 0 \\
\end{bmatrix}
\end{equation*}
% ==============================================================
% ================ MATRICES DIAGONALES ==================
% ==============================================================
\clearpage
\section{Matrices Diagonales}
% ===================================
% ======= DEFINICION ============
% ===================================
\subsection{Definición}
Son todas las matrices cuadradas donde cada elemento cumple que:
\begin{equation*}
[A]_{i,j} = [A]_{i,j} \cdot \delta(i,j)
\end{equation*}
O más formalmente como cualquier matriz que cumple con que:
\begin{equation*}
\BigBrackets{f(i,j)}_{i, j = 1}^{m, n}
=
\BigBrackets{f(i,j) \cdot \delta(i,j) }_{i, j = 1}^{m, n}
\end{equation*}
\textbf{Es decir es una matriz en la que a cualquier elemento lo puedes multiplicar
por la Delta de Kronecker correspondiente y no se vera afectado}.
Una matriz diagonal tiene el siguiente aspecto:
\begin{equation*}
A_n =
\begin{bmatrix}[cccc]
a_{1,1} & 0 & \dots & 0 \\
0 & a_{2,2} & \dots & 0 \\
\vdots & & & \vdots \\
0 & 0 & \dots & a_{n,n} \\
\end{bmatrix}
\end{equation*}
Notemos que las entradas diagonales de una matriz diagonal pueden ser iguales o cero.
Por ejemplo, la matriz cuadrada nula $0_{n, n}$ es una matriz diagonal.
Es un error común pensar que las entradas diagonales de una matriz diagonal deben
ser distintas de cero.
% ===================================
% ======= PROPIEDADES ============
% ===================================
\clearpage
\subsection{Propiedades}
Sea $diag(a_1, \dots, a_n)$ una forma en la que representamos a una matriz diagonal,
despues de todo, $diag$ tendrá $n$ entradas, por lo tanto representará a una matriz
de $n \times n$ donde $a_1, \dots, a_n$ son las entradas de la diagonal, mientras que
todas las demas entradas son cero.
\begin{itemize}
\item
$diag(a_1, \dots, a_n) + (b_1, \dots, b_n) = (a_1+b_1, \dots, a_n+b_n)$
\item
$diag(a_1, \dots, a_n)(b_1, \dots, b_n) = (a_1b_1, \dots, a_nb_n)$
\item
La matriz $diag(a_1, \dots, a_n)$ es invertible si y solo si todas las entradas,
es decir $a_1, \dots, a_n$ son diferentes de cero.
\end{itemize}
% ===================================
% === MATRICES TRIANGULARES =====
% ===================================
\clearpage
\section{Matrices Triangulares Superiores}
Son aquellas matrices de $A \in M_{n \times n}(\GenericField)$ donde se cumple que:
\begin{equation*}
\BigBrackets{f(i,j)}_{i, j = 1}^{n, n}
=
\Brackets{
\begin{cases}
f(i,j) \MiniSpace& \text{ si } i \leq j \\
0 \MiniSpace& \text{ si } i > j
\end{cases}
}_{i, j = 1}^{m, n}
\end{equation*}
Es decir $\forall i, j \in \Set{1, \dots, n} \MegaSpace i > j \implies \Space [A]_{i, j}$
\vspace{1em}
Notemos que en una matriz triangular superior algunos (hasta todos) de los elementos
por encima de la diagonal principal o en la diagonal principal pueden ser iguales a cero.
Por ejemplo, la matriz nula $0_{n,n}$ es triangular superior. La condición que define
matrices triangulares superiores solo nos dice que todos los elementos por debajo de la
diagonal principal deben cero iguales a cero.
Una matriz triangular superior tiene el siguiente aspecto:
\begin{equation*}
A_n =
\begin{bmatrix}[cccc]
a_{1,1} & a_{1, 2} & \dots & a_{1, n} \\
0 & a_{2,2} & \dots & a_{2, n} \\
\vdots & & & \vdots \\
0 & 0 & \dots & a_{n,n} \\
\end{bmatrix}
\end{equation*}
% ===================================
% ======= PROPIEDADES ============
% ===================================
\clearpage
\subsubsection{Propiedades}
\begin{itemize}
\item
Sea $A, B$ matrices triangulares superiores, entonces $AB$ es también
una matriz triangular superior, donde se tiene que:\\
$[AB]_{i, i} = [A]_{i, i} [B]_{i, i}$
% ======== DEMOSTRACION ========
\begin{SmallIndentation}[1em]
\textbf{Demostración}:
Empecemos por ver que es una matriz diagonal, sea $i > j$ entonces
vamos a demostrar que esa entrada es cero.
\begin{align*}
[AB]_{i, j}
&= \sum_{k = 1}^n [A]_{i, k} [B]_{k, j}
&& \Remember{Definición} \\
&=
\sum_{k = 1}^j [A]_{i, k} [B]_{k, j}
+ \sum_{k = j+1}^{i - 1}[A]_{i, k} [B]_{k, j}
+ \sum_{k = i}^n [A]_{i, k} [B]_{k, j}
&& \Remember{Separamos en 3 sumas} \\
&=
\sum_{k = 1}^j (0) [B]_{k, j}
+ \sum_{k = j+1}^{i - 1}(0) [B]_{k, j}
+ \sum_{k = i}^n [A]_{i, k} [B]_{k, j}
&& \Remember{Siempre $i > k$, por eso $[A]_{i, k = 0}$} \\
&=
\sum_{k = 1}^j (0) [B]_{k, j}
+ \sum_{k = j+1}^{i - 1}(0)(0)
+ \sum_{k = i}^n [A]_{i, k} (0)
&& \Remember{Siempre $k > j$, por eso $[B]_{k, j = 0}$} \\
&= 0
\end{align*}
Ahora veamos que $[AB]_{i, i} = [A]_{i, i} [B]_{i, i}$:
\begin{align*}
[AB]_{i, i}
&= \sum_{k = 1}^n [A]_{i, k} [B]_{k, i}
&& \Remember{Definición} \\
&=
\sum_{k = 1}^{i - 1} [A]_{i, k} [B]_{k, i}
+ [A]_{i, i} [B]_{i, i}
+ \sum_{k = i+1}^n [A]_{i, k} [B]_{k, i}
&& \Remember{Separamos en 3 sumas} \\
&=
\sum_{k = 1}^{i - 1} (0) [B]_{k, i}
+ [A]_{i, i} [B]_{i, i}
+ \sum_{k = i+1}^n [A]_{i, k} [B]_{k, i}
&& \Remember{Ve que $i > k$} \\
&=
\sum_{k = 1}^{i - 1} (0) [B]_{k, i}
+ [A]_{i, i} [B]_{i, i}
+ \sum_{k = i+1}^n [A]_{i, k} (0)
&& \Remember{Ve que $k > i$} \\
&= [A]_{i, i} [B]_{i, i}
&& \Remember{Mira que bonita fórmula}
\end{align*}
\end{SmallIndentation}
\item
Si $A$ es una matriz triangular es invertible entonces $A^{-1}$ también
será invertible.
\end{itemize}
% ===============================================================================
% =================== OPERACIONES CON MATRICES ======================
% ===============================================================================
\clearpage
\chapter{Álgebra Matricial}
% ==============================================
% ========== SUMA DE MATRICES =========
% ==============================================
\clearpage
\section{Suma de Matrices}
Definimos la suma de dos Matrices $A, B \in M_{m \times n}(\GenericField)$
como una relación:
\begin{equation*}