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add typos.toml config and fix typos

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arnaucube
2024-03-16 11:36:56 +01:00
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24 changed files with 51 additions and 26 deletions
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abstract-algebra-charles-pinter-notes.tex
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@@ -97,7 +97,7 @@ Every subgroup of a cyclic group is cyclic.
\begin{definition}[Subgroup]
Let $G$ be a group, and $H$ a non-empty subset of $G$. If
\begin{enumerate}[i.]
\item the idenity $e$ of $G$ is in $H$.
\item the identity $e$ of $G$ is in $H$.
\item $H$ is closed with respect to the operation. Which is for $a, b \in H$, $ab \in H$.
\item $H$ is closed with respect to inverses. Which is for $a \in H$, $a^{-1} \in H$.
\end{enumerate}
@@ -174,7 +174,7 @@ In finite sets, if $f: A \rightarrow B$ is injective then $|A| \leq |B|$, and if
\section{Cosets}
\begin{definition}[Coset]
Let $G$ be a group, and $H$ a subgroup of $G$. For any element $a$ in $G$, the symbol $aH$ denotes the set of all products $ah$, as $a$ remains fixed and $h$ ranges over $H$. $aH$ is caled a \emph{left coset} of $H$ in $G$.
Let $G$ be a group, and $H$ a subgroup of $G$. For any element $a$ in $G$, the symbol $aH$ denotes the set of all products $ah$, as $a$ remains fixed and $h$ ranges over $H$. $aH$ is called a \emph{left coset} of $H$ in $G$.
\\
In similar fashion, $Ha$ denotes the set of all products $ha$, as $a$ remains fixed an $h$ ranges over $H$. $Ha$ is called a \emph{right coset} of $H$ in $G$.
\end{definition}
@@ -306,7 +306,7 @@ Quotient group construction is useful as a way of actually manufacturing all the
\end{definition}
\begin{definition}[Unity]
A ring does not necessarily have a neutral element for multiplication. If there is in $A$ a neutral element for mulitplication, it is called the \emph{unity} of $A$, and is denoted by the symbol $1$.
A ring does not necessarily have a neutral element for multiplication. If there is in $A$ a neutral element for multiplication, it is called the \emph{unity} of $A$, and is denoted by the symbol $1$.
\\
If $A$ has a unity, we call $A$ a \emph{ring with unity}.
\end{definition}
@@ -531,7 +531,7 @@ Let $a(x) \in F[x]$ be a polynomial of degree $n$. There is an extension field $
The set of all the linear combinations of $\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$ is a \emph{subspace of} $V$.
\end{definition}
\begin{definition}[Linear dependancy]
\begin{definition}[Linear dependency]
Let $S = \{$\overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n}$\}$ be a set of distinct vectors in a vector space $V$. $S$ is said to be \emph{linearly dependent} if there are scalars $k_1, \ldots, k_n$, not all zero, such that $k_1 \overrightarrow{a_1} + k_2 \overrightarrow{a_2} + \cdots + k_n \overrightarrow{a_n} = 0$.
Which is equivalent to saying that at least one of the vectors in $S$ is a linear combination of the others.