From 33101973d40fdfe2ebd48dc574c290d4ffa9ff37 Mon Sep 17 00:00:00 2001 From: Alex Clarke Date: Mon, 19 Feb 2024 16:40:39 -0700 Subject: [PATCH] Wrote out Chapter 1 theorems and definitions --- Linear-Algebra-Theorems-and-Definitions.pdf | Bin 160192 -> 206612 bytes chapter-1/bases-and-dimension.tex | 106 +++++++++++ chapter-1/introduction.tex | 10 ++ ...ations-and-systems-of-linear-equations.tex | 20 +++ ...ear-dependence-and-linear-independence.tex | 42 +++++ .../maximal-linearly-independent-subsets.tex | 37 ++++ chapter-1/subspaces.tex | 59 ++++++ chapter-1/vector-spaces.tex | 169 ++++++++++++++++++ init.sty | 5 +- 9 files changed, 447 insertions(+), 1 deletion(-) diff --git a/Linear-Algebra-Theorems-and-Definitions.pdf b/Linear-Algebra-Theorems-and-Definitions.pdf index 77a8075a5c2841dccb0c0aacbcf73de03f08f427..dc6d4dbfb6276150b707f8e17951362c2b41af06 100644 GIT binary patch delta 160496 zcmZU)b8z5K&^{P*W81d5v9qylZft*&Y;4=MZQI${wl?;C@Atk{cXd}ae>^o+-8EA) 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zGHb6NBh;-tfbN!TX6j$66;=&R3fhzDdnp zE*u)UjkHdz))KoJph1DCp<}4x)+o25zaqk*=dVx?3CU@z$!NP|H7}Pz8wa^H=sMoQ z>hgENbLc6Zglu4~wQDuNzj!cpm&F7OYhWJB|IdKcd_FaNFZNW?4`ew#2d16<1jU1A RV&mlE j$. +\end{definition} \ No newline at end of file diff --git a/chapter-1/vector-spaces.tex b/chapter-1/vector-spaces.tex index af0fad4..edfb87e 100644 --- a/chapter-1/vector-spaces.tex +++ b/chapter-1/vector-spaces.tex @@ -1 +1,170 @@ \section{Vector Spaces} + +\begin{definition} + \hfill\\ + A \textbf{vector space} (or \textbf{linear space}) $V$ over a field $\F$ consists of a set on which two operations (called \textbf{addition} and \textbf{scalar multiplication}, respectively) are defined so that for each pair of elements $x$ and $y$ in $V$ there is a unique element $a$ in $\F$ and each element $x$ in $V$ there is a unique element $ax$ in $V$, such that the following conditions hold: + + \begin{description} + \item[(VS 1)] For all $x, y$ in $V$, $x + y = y + x$ (commutativity of addition). + \item[(VS 2)] For all $x, y$ in $V$, $(x + y) + z = x + (y + z)$ (associativity of addition). + \item[(VS 3)] There exists an element in $V$ denoted by $0$ such that $x + 0 = x$ for each $x$ in $V$. + \item[(VS 4)] For each element $x$ in $V$ there exists an element $y$ in $V$ such that $x + y = 0$. + \item[(VS 5)] For each element $x$ in $V$, $1x=x$. + \item[(VS 6)] For each pair of elements $a, b$ in $\F$ and each element $x$ in $V$, $(ab)x = a(bx)$. + \item[(VS 7)] For each element $a$ in $\F$ and each pair of elements $x, y$ in $V$, $a(x + y) = ax + ay$. + \item[(VS 8)] For each pair of elements $a, b$ in $\F$ and each element $x$ in $V$, $(a + b)x = ax + bx$. + \end{description} + + The elements $x + y$ and $ax$ are called the \textbf{sum} of $x$ and $y$ and the \textbf{product} of $a$ and $x$, respectively.\\ + + The elements of the field $\F$ are called \textbf{scalars} and the elements of the vector space $V$ are called \textbf{vectors}.\\ + + \textbf{Note:} The reader should not confuse this use of the word "vector" with the physical entity discussed in section 1.1: the word "vector" is now being used to describe any element of a vector space. +\end{definition} + +\begin{definition} + \hfill\\ + An object of the form $(a_1, a_2, \dots, a_n)$, where the entries $a_1, a_2, \dots, a_n$ are elements of a field $\F$, is called an \textbf{\textit{n}-tuple} with entries from $\F$. The elements $a_1, a_2, \dots, a_n$ are called the \textbf{entries} or \textbf{components} of the $n$-tuple. Two $n$-tuples $(a_1, a_2, \dots, a_n)$ and $(b_1, b_2, \dots, b_n)$ with entries from a field $\F$ are called \textbf{equal} if $a_i = b_i$ for $i=1, 2, \dots, n$. +\end{definition} + +\begin{definition} + \hfill\\ + Vectors in $\F^n$ may be written as \textbf{column vectors} + + \[\begin{pmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{pmatrix}\] + rather than as \textbf{row vectors} $(a_1, a_2, \dots, a_n)$. Since a 1-tuple whose only entry is from $\F$ can be regarded as an element of $\F$, we usually write $\F$ rather than $\F^1$ for the vector space of 1-tuples with entry from $\F$. +\end{definition} + +\begin{definition} + \hfill\\ + An $m \times n$ \textbf{matrix} with entries from a field $\F$ is a rectangular array of the form + + \[\begin{pmatrix} + a_{11} & a_{12} & \dots &a_{1n} \\ + a_{21} & a_{22} & \dots & a_{2n} \\ + \vdots & \vdots & & \vdots \\ + a_{m1} & a_{m2} & \dots & a_{mn} + \end{pmatrix},\] + where each entry $a_{ij}\ (1 \leq i \leq m,\ 1 \leq j \leq n)$ is an element of $\F$. We call the entries $a_{ij}$ with $i=j$ the \textbf{diagonal entries} of the matrix. The entries $a_{i1}, a_{i2}, \dots, a_{in}$ compose the \textbf{\textit{i}th row} of the matrix, and the entries $a_{1j}, a_{2j}, \dots, a_{mj}$ compose the \textbf{\textit{j}th column} of the matrix. The rows of the preceding matrix are regarded as vectors in $\F^n$, and the columns are regarded as vectors in $\F^m$. The $m \times n$ matrix in which each entry equals zero is called the \textbf{zero matrix} and is denoted by $O$.\\ + + In this book, we denote matrices by capital italic letters (e.g. $A$, $B$, and $C$), and we denote the entry of a matrix $A$ that lies in row $i$ and column $j$ by $A_{ij}$. In addition, if the number of rows and columns of a matrix are equal, the matrix is called \textbf{square}. + + Two $m \times n$ matrices $A$ and $B$ are called \textbf{equal} if all their corresponding entries are equal, that is, if $A_{ij} = B_{ij}$ for $1 \leq i \leq m$ and $1 \leq j \leq n$. +\end{definition} + +\begin{definition} + \hfill\\ + The set of all $m \times n$ matrices with entries from a field $\F$ is a vector space which we denote by $M_{m \times n}(\F)$, with the following operations of \textbf{matrix addition} and \textbf{scalar multiplication}: For $A, B \in M_{m \times n}(\F)$ and $c \in \F$, + + \[(A + B)_{ij} = A_{ij} + B_{ij}\ \ \ \text{and}\ \ \ (cA)_{ij} = cA_{ij}\] + for $1 \leq i \leq m$ and $1 \leq j \leq n$. +\end{definition} + +\begin{definition} + \hfill\\ + Let $S$ be any nonempty set and $\F$ be any field, and let $\mathcal{F}(S, \F)$ denote the set of all functions from $S$ to $\F$. Two functions $f$ and $g$ in $\mathcal{F}(S, \F)$ are called \textbf{equal} if $f(s) = g(s)$ for each $s \in S$. The set $\mathcal{F}(S, \F)$ is a vector space with the operations of addition and scalar multiplication defined for $f,g \in \mathcal{F}(S, \F)$ and $c \in \F$ defined by + + \[(f + g)(s) = f(s) + g(s)\ \ \ \text{and}\ \ \ (cf)(s) = c[f(s)]\] + for each $s \in S$. Note that these are the familiar operations of addition and scalar multiplication for functions used in algebra and calculus. +\end{definition} + +\begin{definition} + \hfill\\ + A \textbf{polynomial} with coefficients from a field $\F$ is an expression of the form + + \[f(x)=a_nx^n + a_{n-1}x^{n-1}+\dots+a_1x+a_0,\] + + where $n$ is a nonnegative integer and each $a_k$, called the \textbf{coefficient} of $x^k$, is in $\F$. If $f(x)=0$, that is, if $a_n = a_{n-1} = \dots = a_0 = 0$, then $f(x)$ is called the \textbf{zero polynomial} and, for convenience, its degree is defined to be $-1$; otherwise, the \textbf{degree} of a polynomial is defined to be the largest exponent of $x$ that appears in the representation + + \[f(x)=a_nx^n + a_{n-1}x^{n-1}+\dots+a_1x+a_0\] + + with a nonzero coefficient. Note that the polynomials of degree zero may be written in the form $f(x) = c$ for some nonzero scalar $c$. Two polynomials, + + \[f(x)=a_nx^n + a_{n-1}x^{n-1}+\dots+a_1x+a_0\] + + and + + \[g(x)=b_mx^m + b_{m-1}x^{m-1}+\dots+b_1x+b_0,\] + + are called \textbf{equal} if $m=n$ and $a_i = b_i$ for $i=1, 2, \dots, n$. +\end{definition} + +\begin{definition} + \hfill\\ + Let $\F$ be any field. A \textbf{sequence} in $\F$ is a function $\sigma$ from the positive integers into $\F$. In this book, the sequence $\sigma$ such that $\sigma(n) = a_n$ for $n=1, 2, \dots$ is denoted $\{a_n\}$. Let $V$ consist of all sequences $\{a_n\}$ in $\F$ that have only a finite number of nonzero terms $a_n$. If $\{a_n\}$ and $\{b_n\}$ are in $V$ and $t \in \F$, define + + \[\{a_n\} + \{b_n\} = \{a_n + b_n\}\ \ \ \text{and}\ \ \ t\{a_n\} = \{ta_n\}\] +\end{definition} + +\begin{theorem}[\textbf{Cancellation Law for Vector Addition}] + \hfill\\ + If $x, y$ and $z$ are vectors in a vector space $V$ such that $x + z = y + z$, then $x = y$. +\end{theorem} + +\begin{corollary} + \hfill\\ + The vector $0$ described in (VS 3) is unique. +\end{corollary} + +\begin{corollary} + \hfill\\ + The vector $y$ described in (VS 4) is unique. +\end{corollary} + +\begin{definition} + \hfill\\ + The vector $0$ in (VS 3) is called the \textbf{zero vector} of $V$, and the vector $y$ in (VS 4) (that is, the unique vector such that $x + y = 0$) is called the \textbf{additive inverse} of $x$ and is denoted by $-x$. +\end{definition} + +\begin{theorem} + \hfill\\ + In any vector space $V$, the following statements are true: + \begin{enumerate} + \item $0x = 0$ for each $x \in V$. + \item $(-a)x = -(ax) = a(-x)$ for each $a \in \F$ and each $x \in V$. + \item $a0 = 0$ for each $a \in \F$. + \end{enumerate} +\end{theorem} + +\begin{definition} + \hfill\\ + Let $V=\{0\}$ consist of a single vector $0$ and define $0 + 0 = 0$ and $c0 = 0$ for each scalar $c \in \F$. Then $V$ is called the \textbf{zero vector space}. +\end{definition} + +\begin{definition} + \hfill\\ + A real-valued function $f$ defined on the real line is called an \textbf{even function} if $f(-t) = f(t)$ for each real number $t$, and is called an \textbf{odd function} if $f(-t) = -f(t)$ for each real number $t$. +\end{definition} + +\begin{definition} + \hfill\\ + If $S_1$ and $S_2$ are nonempty subsets of a vector space $V$, then the \textbf{sum} of $S_1$ and $S_2$, denoted $S_1 + S_2$, is the set $\{x + y\ |\ x \in S_1,\ \text{and}\ y \in S_2\}$. +\end{definition} + +\begin{definition} + \hfill\\ + A vector space $V$ is called the \textbf{direct sum} of $W_1$ and $W_2$ if $W_1$ and $W_2$ are subspaces of $V$ such that $W_1 \cap W_2 = \{0\}$ and $W_1 + W_2 = V$. We denote that $V$ is the direct sum of $W_1$ and $W_2$ by writing $V = W_1 \oplus W_2$. +\end{definition} + +\begin{definition} + \hfill\\ + A matrix $M$ is called \textbf{skew-symmetric} if $M^t = -M$. +\end{definition} + +\begin{definition} + \hfill\\ + Let $W$ be a subspace of a vector space $V$ over a field $\F$. For any $v \in V$, the set $\{v\} + W = \{v + w\ |\ w \in W\}$ is called the \textbf{coset of $W$ containing $v$}. It is customary to denote this coset by $v + W$ rather than $\{v\} + W$. +\end{definition} + +\begin{definition} + \hfill\\ + Let $W$ be a subspace of a vector space $V$ over a field $\F$, and let $S := \{v + W\ |\ v \in V\}$ be the set of all cosets of $W$. Then $S$ is called the \textbf{quotient space of $V$ modulo $W$}, and is denoted by $V/W$. Addition and scalar multiplication by the scalars of $\F$ can be defined as follows: + + \[(v_1 + W) + (v_2 + W) = (v_1 + v_2) + W\] + + for all $v_1, v_2 \in V$, and + + \[a(v + W) = av + W\] + + for all $v \in V$ and $a \in \F$. +\end{definition} \ No newline at end of file diff --git a/init.sty b/init.sty index d4dd92c..7ed7d43 100644 --- a/init.sty +++ b/init.sty @@ -23,6 +23,9 @@ \usepackage{enumitem} \usepackage{mathrsfs} +% Spacing functions +\newcommand{\thmfill}{\hfill\\} + % Math Functions \newcommand{\limx}[2]{\displaystyle\lim\limits_{#1 \to #2}} \newcommand{\st}{\ \text{s.t.}\ } @@ -52,7 +55,7 @@ \newcommand{\F}{\mathbb{F}} % Theorem Styles -\declaretheorem[numberwithin=subsection, style=definition]{theorem, definition, notation, lemma, corollary, remark, example} +\declaretheorem[numberwithin=section, style=definition]{theorem, definition, notation, lemma, corollary, remark, example} % Formatting \setlist[enumerate]{font=\bfseries}

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