From e2d88b824883568ab9dadc88f9243c8e5d5f0a43 Mon Sep 17 00:00:00 2001
From: Nao Pross <np@0hm.ch>
Date: Thu, 4 Mar 2021 22:53:53 +0100
Subject: More about fields and vector spaces

---
 FuVar.tex | 68 ++++++++++++++++++++++++++++++++++++++++++---------------------
 1 file changed, 46 insertions(+), 22 deletions(-)

diff --git a/FuVar.tex b/FuVar.tex
index 5d1fdb9..aa4ba54 100644
--- a/FuVar.tex
+++ b/FuVar.tex
@@ -52,6 +52,7 @@
 \newcommand\Qset{\mathbb{Q}}
 \newcommand\Rset{\mathbb{R}}
 \newcommand\Cset{\mathbb{C}}
+\newcommand\T{\mathrm{T}}
 
 %% Theorems
 \newtheoremstyle{fuvarzf} % name of the style to be used
@@ -70,14 +71,9 @@
 \newtheorem{lemma}{Lemma}
 
 \setlist[description]{
-  align = right, labelwidth = 2cm, leftmargin = !,
   format = { \normalfont\itshape }
 }
 
-\setlist[itemize]{
-  align = right, labelwidth = 5mm, leftmargin = !
-}
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % Document
 
@@ -87,20 +83,19 @@
 \tableofcontents
 
 \section{Fields and vector spaces}
+
 \begin{definition}[Field]
   A field is a set \(F\) with two binary operators \(+\) and \(\cdot\) that map
   \(F\times F \to F\) and follow the \emph{field axioms} listed below. We let 
-  \(a, (-a), b, b^{-1} \in F\) and \(\star\) stand for either \(\cdot\) or \(+\).
-  \begin{description}[leftmargin=2.5cm]
-    \item[Associativity] \((a \star b) \star c = a \star (b \star c)\)
-    \item[Commutativity] \(a \star b = b \star a\) 
-    \item[Identities] \(0 + a = a\) and \(1\cdot a = a\)
-    \item[Inverses]
-      \begin{itemize}
-        \item \(a + (-a) = 0\) and 
-        \item \(b \cdot b^{-1} = 1\) where \(b \neq 0\)
-      \end{itemize}
-    \item[Distributivity] \(a \cdot (b + c) = a\cdot b + a \cdot c\)
+  \(a, b \in F\) and \(\star\) stands for \(\cdot\) or \(+\).
+  \begin{description}
+    \item[Associativity:] \((a \star b) \star c = a \star (b \star c)\)
+    \item[Commutativity:] \(a \star b = b \star a\) 
+    \item[Identities:] \(0 + a = a\) and \(1\cdot a = a\)
+    \item[Inverses:]
+        \(a + (-a) = 0\) and 
+        \(b \cdot b^{-1} = 1\) iff \(b \neq 0\)
+    \item[Distributivity:] \(a \cdot (b + c) = a\cdot b + a \cdot c\)
   \end{description}
 \end{definition}
 
@@ -109,21 +104,50 @@
 \end{theorem}
 
 \begin{definition}[Vector space]
-  A vector \(V\) space over a field \(F\)
+  A vector space \(U\) over a field \(F\) is a set of objects called
+  \emph{vectors} equipped with two operations: \emph{addition} 
+  \(+: U \times U \to U\) and \emph{scalar multiplication} 
+  \(\cdot: F\times U \to U\), that respect the following axioms.
+  Let \(\vec{u}, \vec{v}, \vec{w} \in U\) and \(a, b \in F\).
+  \begin{description}
+    \item[Additive associativity:] \((\vec{u} + \vec{v}) + \vec{w} 
+      = \vec{u} + (\vec{v} + \vec{w})\)
+    \item[Additive commutativity:] \(\vec{u} + \vec{v} = \vec{v} + \vec{u}\)
+    \item[Identities:] There is an element 
+      \(\vec{0} \in U : \vec{u} + \vec{0} = \vec{u}\) 
+      and \(1 \in F : 1 \cdot \vec{u} = \vec{u}\)
+    \item[Additive inverse:] \(\vec{u} + (\vec{-u}) = 0\)
+    \item[Compatibility of multiplication] 
+      \(a\cdot (b \cdot \vec{u}) = (a\cdot b) \cdot \vec{u}\)
+    \item[Distributivity:]
+      \((a + b) \cdot \vec{u} = a\cdot\vec{u} + b\cdot\vec{u}\) and conversely
+      \(a \cdot (\vec{u} + \vec{v}) = a\cdot\vec{u} + a\cdot\vec{v}\)
+  \end{description}
+  And of course elements in \(F\) follow the field axioms.
 \end{definition}
 
 \begin{theorem}
-  \(\Rset^n\) is a vector space.
+  \(\Rset^n = \Rset\times\cdots\times\Rset\) is a vector space.
 \end{theorem}
 
 \begin{definition}[Row and column vectors]
+  Although there is virtually no difference between the two, we need two type
+  of \(n\)-tuples that satisfy the vector space axioms. \emph{Row} vectors are
+  written horizontally and \emph{column} vectors vertically.
+\end{definition}
+
+\begin{definition}[Transposition]
+  Let \(\vec{u} \in \Rset^n\) be a row vector. The \emph{transpose} of
+  \(\vec{u}\) denoted with \(\vec{u}^\T\) is column vector with the same
+  components. Conversely if \(\vec{v}\) is a column vector then \(\vec{v}^\T\)
+  is a row vector.
 \end{definition}
 
-\section{Scalar Fields}
+\section{Scalar fields}
 
-\begin{definition}[Scalar Field]
-  We call a function \(f\) a \emph{scalar field} when it maps values from
-  \(\Rset^n \to \Rset\).
+\begin{definition}[Scalar field]
+  Confusingly we call a function \(f: \Rset^n \to \Rset\) a \emph{scalar
+  field}, but this is unrelated to the previously defined field.
 \end{definition}
 
 \begin{definition}[Partial derivative of a scalar field]
-- 
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