Added subsections when they appear, added all of the appendices, and finished the packet
This commit is contained in:
@@ -0,0 +1,6 @@
|
||||
\chapter{Appendices}
|
||||
\subimport{./}{sets.tex}
|
||||
\subimport{./}{functions.tex}
|
||||
\subimport{./}{fields.tex}
|
||||
\subimport{./}{complex-numbers.tex}
|
||||
\subimport{./}{polynomials.tex}
|
||||
@@ -0,0 +1,83 @@
|
||||
\begin{alphasection}
|
||||
\setcounter{alphasect}{3}
|
||||
\section{Complex Numbers}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A \textbf{complex number} is an expression of the form $z = a + bi$, where $a$ and $b$ are real numbers called the \textbf{real part} and the \textbf{imaginary part} of $z$, respectively.
|
||||
|
||||
The \textbf{sum} and \textbf{product} of two complex numbers $z = a + bi$ and $w = c+di$ (where $a$, $b$, $c$, and $d$ are real numbers) are defined, respectively, as follows:
|
||||
|
||||
\[z+w = (a + bi) + (c+di) = (a+c) + (b+d)i\]
|
||||
|
||||
and
|
||||
|
||||
\[zw = (a+bi)(c+di) = (ac-bd)+(bc+ad)i\]
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Any complex number of the form $bi=0 + bi$, where $b$ is a nonzero real number, is called \textbf{imaginary}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
The set of complex numbers with the operations of addition and multiplication previously defined is a field.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
The (\textbf{complex}) \textbf{conjugate} of a complex number $a+bi$ is the complex number $a-bi$. We denote the conjugate of a complex number $z$ by $\overline{z}$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $z$ and $w$ be complex numbers. Then the following statements are true.
|
||||
|
||||
\begin{enumerate}
|
||||
\item $\overline{\overline{z}} = z$.
|
||||
\item $\overline{(z+ w)} = \overline{z}+ \overline{w}$.
|
||||
\item $\overline{zw} = \overline{z}\cdot\overline{w}$.
|
||||
\item $\overline{(\frac{z}{w})} = \frac{\overline{z}}{\overline{w}}$ if $w \neq 0$.
|
||||
\item $z$ is a real number if and only if $\overline{z} = z$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Let $z = a + bi$, where $a,b \in \R$. The \textbf{absolute value} (or \textbf{modulus}) of $z$ is the real number $\sqrt{a^2 + b^2}$. We denote the absolute value of $z$ as $|z|$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $z$ and $w$ denote any two complex numbers. Then the following statements are true.
|
||||
|
||||
\begin{enumerate}
|
||||
\item $|zw| = |z| \cdot |w|$.
|
||||
\item $\abs{\frac{z}{w}} = \frac{|z|}{|w|}$ if $w \neq 0$.
|
||||
\item $|z + w| \leq |z| + |w|$.
|
||||
\item $|z| - |w| \leq |z + w|$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Notice that, as in $\R^2$, there are two axes, the \textbf{real axis} and the \textbf{imaginary axis}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[\textbf{The Fundamental Theorem of Algebra}]
|
||||
\hfill\\
|
||||
Suppose that $p(z) = a_nz^n + a_{n-1}z^{n-1} + \dots + a_1z + a_0$ is a polynomial in $P(\C)$ degree $n \geq 1$. Then $p(z)$ has a zero.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
If $p(z) = a_nz^n + a_{n-1}z^{n-1} + \dots + a_1z + a_0$ is a polynomial of degree $n \geq 1$ with complex coefficients, then there exists complex numbers $c_1, c_2, \dots, c_n$ (not necessarily distinct) such that
|
||||
|
||||
\[p(z) = a_n(z-c_1)(z-c_2)\dots(z-c_n).\]
|
||||
\end{corollary}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A field is called \textbf{algebraically closed} if it has the property that every polynomial of positive degree 1. Thus the preceding corollary asserts that the field of complex numbers is algebraically closed.
|
||||
\end{definition}
|
||||
\end{alphasection}
|
||||
@@ -0,0 +1,65 @@
|
||||
\begin{alphasection}
|
||||
\setcounter{alphasect}{2}
|
||||
\section{Fields}
|
||||
|
||||
\begin{definition}
|
||||
A field $\F$ is a set on which two operations $+$ and $\cdot$ (called \textbf{addition} and \textbf{multiplication}, respectively) are defined so that, for each pair of elements $x,y \in \F$, there are unique elements $x+y$ and $x \cdot y$ in $\F$ for which the following conditions hold for all elements $a,b,c \in \F$.
|
||||
|
||||
\begin{enumerate}
|
||||
\item[(F 1)] $a + b = b + a$ and $a\cdot b = b \cdot a$\\
|
||||
(commutativity of addition and multiplication)
|
||||
\item[(F 2)] $(a + b) + c = a + (b + c)$ and $(a \cdot b)\cdot c = a \cdot (b \cdot c)$\\
|
||||
(associativity of addition and multiplication)
|
||||
\item[(F 3)] There exist distinct elements $0$ and $1$ in $\F$ such that
|
||||
|
||||
\[0+a = a\ \ \ \ \text{and}\ \ \ \ 1\cdot a = a\]
|
||||
(existence of identity elements for addition and multiplication)
|
||||
|
||||
\item[(F 4)] For each element $a$ in $\F$ and each nonzero element $b$ in $\F$, there exists elements $c$ and $d$ in $\F$ such that
|
||||
|
||||
\[a+c = 0\ \ \ \ \text{and}\ \ \ \ b\cdot d = 1\]
|
||||
(existence of inverses for addition and multiplication)
|
||||
|
||||
\item[(F 5)] $a \cdot(b + c) = a\cdot b + a \cdot c$\\
|
||||
(distributivity of multiplication over addition)
|
||||
\end{enumerate}
|
||||
|
||||
The elements $x + y$ and $x \cdot y$ are called the \textbf{sum} and \textbf{product}, respectively, of $x$ and $y$. The elements $0$ (read ``\textbf{zero}") and 1 (read ``\textbf{one}") mentioned in (F 3) are called \textbf{identity elements} for addition and multiplication, respectively, and the elements $c$ and $d$ referred to in (F 4) are called an \textbf{additive inverse} for $a$ and a \textbf{multiplicative inverse} for $b$, respectively.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[\textbf{Cancellation Laws}]
|
||||
\hfill\\
|
||||
For arbitrary elements $a$, $b$, and $c$ in a field, the following statements are true.
|
||||
|
||||
\begin{enumerate}
|
||||
\item If $a + b = c + b$, then $a=c$.
|
||||
\item If $a \cdot b = \cdot b$ and $b\ neq 0$, then $a=c$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
\hfill\\
|
||||
The elements $0$ and $1$ mentioned in (F 3), and the elements $c$ and $d$ mentioned in (F 4), are unique.
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $a$ and $b$ be arbitrary elements of a field. Then each of the following statements are true.
|
||||
|
||||
\begin{enumerate}
|
||||
\item $a \cdot 0 = 0$.
|
||||
\item $(-a)\cdot b = a \cdot(-b) = -(a \cdot b)$.
|
||||
\item $(-a) \cdot (-b) = a \cdot b$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
\hfill\\
|
||||
The additive identity of a field has no multiplicative inverse.
|
||||
\end{corollary}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
In an arbitrary field $\F$, the smallest positive integer $p$ for which a sum of $p$ 1's equals 0 is called the \textbf{characteristic} of $\F$; if no such positive integer exists, then $\F$ is said to have \textbf{characteristic zero}.
|
||||
\end{definition}
|
||||
\end{alphasection}
|
||||
@@ -0,0 +1,36 @@
|
||||
\begin{alphasection}
|
||||
\setcounter{alphasect}{1}
|
||||
\section{Functions}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
If $A$ and $B$ are sets, then a \textbf{function} $f$ from $A$ to $B$, written $f: A \to B$, is a rule that associates to each element of $x$ in $A$ a unique element denoted $f(x)$ in $B$.\\
|
||||
|
||||
The element $f(x)$ is called the \textbf{image} of $x$ (under $f$), and $x$ is called a \textbf{preimage} of $f(x)$ (under $f$).\\
|
||||
|
||||
If $f: A \to B$, then $A$ is called the \textbf{domain} of $f$, $B$ is called the \textbf{codomain} of $f$, and the set $\{f(x) : x \in A\}$ is called the \textbf{range} of $f$.\\
|
||||
|
||||
Two functions $f: A \to B$ and $g: A \to B$ are \textbf{equal}, written $f=g$, if $f(x) = g(x)$ for all $x \in A$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Functions such that each element of the range has a unique preimage are called \textbf{one-to-one}; that is, $f: A \to B$ is one-to-one if $f(x) = f(y)$ implies $x=y$ or, equivalently, if $x \neq y$ implies $f(x) \neq f(y)$.\\
|
||||
|
||||
If $f: A \to B$ is a function with range $B$, that is, if $f(A) = B$, then $f$ is called \textbf{onto}. So $f$ is onto if and only if the range of $f$ equals the codomain of $f$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Let $f: A \to B$ be a function and $S \subseteq A$. Then a function $f_S: S \to B$, called the \textbf{restriction} of $f$ to $S$, can be formed by defining $f_S(x) = f(x)$ for each $x \in S$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A function $f: A \to B$ is said to be \textbf{invertible} if there exists a function $g: B \to A$ such that $(f \circ g)(y) = y$ for all $y \in B$ and $(g \circ f)(x)=x$ for all $x \in A$.
|
||||
|
||||
If such a function $g$ exists, then it is unique and is called the \textbf{inverse} of $f$. We denote the inverse of $f$ (when it exists) by $f^{-1}$.
|
||||
|
||||
It can be shown that $f$ is invertible if and only if $f$ is both one-to-one and onto.
|
||||
\end{definition}
|
||||
\end{alphasection}
|
||||
@@ -0,0 +1,133 @@
|
||||
\begin{alphasection}
|
||||
\setcounter{alphasect}{4}
|
||||
\section{Polynomials}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A polynomial $f(x)$ \textbf{divides} a polynomial $g(x)$ if there exists a polynomial $q(x)$ such that $g(x) = f(x)q(x)$.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}[\textbf{The Division Algorithm for Polynomials}]\label{Theorem 8.7}
|
||||
Let $f(x)$ be a polynomial of degree $n$, and let $g(x)$ be a polynomial of degree $m \geq 0$. Then there exists unique polynomials $q(x)$ and $r(x)$ such that
|
||||
|
||||
\[f(x) = q(x)g(x)+r(x),\]
|
||||
|
||||
where the degree of $r(x)$ is less than $x$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
In the context of \autoref{Theorem 8.7}, we call $q(x)$ and $r(x)$ the \textbf{quotient} and \textbf{remainder}, respectively, for the division of $f(x)$ by $g(x)$.
|
||||
\end{definition}
|
||||
|
||||
\begin{corollary}
|
||||
\hfill\\
|
||||
Let $f(x)$ be a polynomial of positive degree, and let $a \in \F$. Then $f(a) = 0$ if and only if $x-a$ divides $f(x)$.
|
||||
\end{corollary}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
For any polynomial $f(x)$ with coefficients from a field $\F$, an element $a \in \F$ is called a \textbf{zero} of $f(x)$ if $f(a) =0$. With this terminology, the preceding corollary states that $a$ is a zero of $f(x)$ if and only if $x-a$ divides $f(x)$.
|
||||
\end{definition}
|
||||
|
||||
\begin{corollary}
|
||||
\hfill\\
|
||||
Any polynomial of degree $n \geq 1$ has at most $n$ distinct zeros.
|
||||
\end{corollary}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Two nonzero polynomials are called \textbf{relatively prime} if no polynomial of positive degree divides each of them.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
If $f_1(x)$ and $f_2(x)$ are relatively prime polynomials, there exist polynomials $q_1(x)$ and $q_2(x)$ such that
|
||||
|
||||
\[q_1(x)f_1(x) + q_2(x)f_2(x) = 1,\]
|
||||
|
||||
where $1$ denoted the constant polynomial with value $1$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Let
|
||||
|
||||
\[f(x) = a_0 + a_1(x) + \dots + a_nx^n\]
|
||||
|
||||
be a polynomial with coefficients from a field $\F$. If $T$ is a linear operator on a vector space $V$ over $\F$, we define
|
||||
|
||||
\[f(T) = a_0I + a_1T + \dots + a_nT^n.\]
|
||||
|
||||
Similarly, if $A$ is an $n \times n$ matrix with entries from $\F$, we define
|
||||
|
||||
\[f(A) = a_0I+ a_1A + \dots + a_nA^n.\]
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $f(x)$ be a polynomial with coefficients from a field $\F$, and let $T$ be a linear operator on a vector space $V$ over $\F$. Then the following statements are true.
|
||||
|
||||
\begin{enumerate}
|
||||
\item $f(T)$ is a linear operator on $V$.
|
||||
\item If $\beta$ is a finite ordered basis for $V$ and $A=[T]_\beta$, then $[f(T)]_\beta = f(A)$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $T$ be a linear operator on a vector space $V$ over a field $\F$, and let $A$ be a square matrix with entries from $\F$. Then, for any polynomials $f_1(x)$ and $f_2(x)$ with coefficients $\F$,
|
||||
|
||||
\begin{enumerate}
|
||||
\item $f_1(T)f_2(T) = f_2(T)f_1(T)$
|
||||
\item $f_1(A)f_2(A) = f_2(A)f_1(A)$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $T$ be a linear operator on a vector space $V$ over a field $\F$, and let $A$ be an $n \times n$ matrix with entries from $\F$. If $f_1(x)$ and $f_2(x)$ are relatively prime polynomials with entries from $\F$, then there exist polynomials $q_1(x)$ and $q_2(x)$ with entries from $\F$ such that
|
||||
|
||||
\begin{enumerate}
|
||||
\item $q_1(T)f_1(T) + q_2(T)f_2(T) = I$
|
||||
\item $q_1(A)f_1(A) + q_2(A)f_2(A) = I$.
|
||||
\end{enumerate}
|
||||
\end{theorem}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A polynomial $f(x)$ with coefficients from a field $\F$ is called \textbf{monic} if its leading coefficient is 1. If $f(x)$ has positive degree and cannot be expressed as a product of polynomials with coefficients from $\F$ each having positive degree, then $f(x)$ is called \textbf{irreducible}.
|
||||
\end{definition}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $\phi(x)$ and $f(x)$ be polynomials. If $\phi(x)$ is irreducible and $\phi(x)$ does not divide $f(x)$, then $\phi(x)$ and $f(x)$ are relatively prime.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Any two distinct irreducible monic polynomials are relatively prime.
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $f(x)$, $g(x)$, and $\phi(x)$ be polynomials. If $\phi(x)$ is irreducible and divides the product $f(x)g(x)$, then $\phi(x)$ divides $f(x)$ or $\phi(x)$ divides $g(x)$.
|
||||
\end{theorem}
|
||||
|
||||
\begin{corollary}
|
||||
\hfill\\
|
||||
Let $\phi(x),\phi_1(x)\phi_2(x), \dots, \phi_n(x)$ be irreducible monic polynomials. If $\phi(x)$ divides the product $\phi_1(x) \phi_2(x) \dots \phi_n(x)$, then $\phi(x) = \phi_i(x)$ for some $i$ ($i = 1, 2, \dots n$).
|
||||
\end{corollary}
|
||||
|
||||
\begin{theorem}[\textbf{Unique Factorization Theorem for Polynomials}]
|
||||
\hfil\\
|
||||
For any polynomial $f(x)$ of positive degree, there exist a unique constant $c$; unique distinct irreducible monic polynomials $\phi_1(x),\phi_2(x), \dots, \phi_n(x)$; and unique positive integers $n_1, n_2, \dots, n_k$ such that
|
||||
|
||||
\[f(x) = c[\phi_1(x)]^{n_1} [\phi_2(x)]^{n_2} \dots [\phi_k(x)]^{n_k}.\]
|
||||
\end{theorem}
|
||||
|
||||
\begin{theorem}
|
||||
\hfill\\
|
||||
Let $f(x)$ and $g(x)$ be polynomials with coefficients from an infinite field $\F$. If $f(a)= g(a)$ for all $a \in \F$, then $f(x)$ and $g(x)$ are equal.
|
||||
\end{theorem}
|
||||
\end{alphasection}
|
||||
@@ -0,0 +1,68 @@
|
||||
\begin{alphasection}
|
||||
\section{Sets}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A \textbf{set} is a collection of objects, called \textbf{elements} of the set. If $x$ is an element of the set $A$, then we write $x \in A$; otherwise, we write $x \notin A$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Two sets $A$ and $B$ are called \textbf{equal}, written $A = B$, if they contain exactly the same elements.
|
||||
\end{definition}
|
||||
|
||||
\begin{remark}
|
||||
\hfill\\
|
||||
Sets may be described in one of two ways
|
||||
|
||||
\begin{enumerate}
|
||||
\item By listing the elements of the set between set braces $\{$ $\}$.
|
||||
\item By describing the elements of the set in terms of some characteristic property.
|
||||
\end{enumerate}
|
||||
\end{remark}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A set $B$ is called a \textbf{subset} of $A$, written $B \supseteq A$ or $A \supseteq B$, if every element of $B$ is an element of $A$. If $B \subseteq A$, and $B \neq A$, then $B$ is called a \textbf{proper subset} of $A$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
The \textbf{empty set}, denoted by $\emptyset$, is the set containing no elements. The empty set is a subset of every set.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
The \textbf{union} of two sets $A$ and $B$, denoted $A \cup B$, is the set of elements that are in $A$, or $B$, or both; that is,
|
||||
|
||||
\[A \cup B = \{x : x \in A\ \text{or}\ B\}.\]
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
The \textbf{intersection} of two sets $A$ and $B$, denoted $A \cap B$, is the set of elements that are in both $A$ and $B$; that is,
|
||||
|
||||
\[A \cap B = \{x : x \in A\ \text{and}\ B\}.\]
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
Two sets are called \textbf{disjoint} if their intersection equals the empty set.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A \textbf{relation} on a set $A$ is a set $S$ of ordered pairs of elements of $A$ such that $(x,y) \in S$ if and only if $x$ stands in the given relationship to $y$.
|
||||
\end{definition}
|
||||
|
||||
\begin{definition}
|
||||
\hfill\\
|
||||
A relation $S$ on a set $A$ is called an \textbf{equivalence relation} on $A$ if the following three conditions hold:
|
||||
|
||||
\begin{enumerate}
|
||||
\item For each $x \in A$, $x \sim x$ (reflexivity).
|
||||
\item If $x \sim y$, then $y \sim x$ (symmetry).
|
||||
\item If $x \sim y$ and $y \sim z$, then $x \sim z$ (transitivity).
|
||||
\end{enumerate}
|
||||
\end{definition}
|
||||
\end{alphasection}
|
||||
Reference in New Issue
Block a user