minor corrections

sols.tex

@@ -15,7 +15,7 @@
 \usepackage{tabls}
 %TCIDATA{OutputFilter=latex2.dll}
 %TCIDATA{Version=5.50.0.2960}
-%TCIDATA{LastRevised=Wednesday, October 03, 2018 01:01:22}
+%TCIDATA{LastRevised=Thursday, October 04, 2018 09:34:08}
 %TCIDATA{SuppressPackageManagement}
 %TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
 %TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
@@ -187,14 +187,16 @@ \section{\label{chp.intro}Introduction}
 vector spaces), exterior powers, eigenvalues, or of the \textquotedblleft
 universal coefficients\textquotedblright\ trick\footnote{This refers to the
 standard trick used for proving determinant identities (and other polynomial
-identities), in which the entries of a matrix are replaced by indeterminates
-and one then uses the \textquotedblleft genericity\textquotedblright\ of these
-indeterminates to (e.g.) invert the matrix.}. (This means that all proofs are
-done through combinatorics and manipulation of sums -- a rather restrictive
-requirement!) This is a conscious and (to a large extent) aesthetic choice on
-my part, and I do \textbf{not} consider it the best way to learn about
-determinants; but I do regard it as a road worth charting, and these notes are
-my attempt at doing so.
+identities), in which one first replaces the entries of a matrix (or, more
+generally, the variables appearing in the identity) by indeterminates, then
+uses the \textquotedblleft genericity\textquotedblright\ of these
+indeterminates (e.g., to invert the matrix, or to divide by an expression that
+could otherwise be $0$), and finally substitutes the old variables back for
+the indeterminates.}. (This means that all proofs are done through
+combinatorics and manipulation of sums -- a rather restrictive requirement!)
+This is a conscious and (to a large extent) aesthetic choice on my part, and I
+do \textbf{not} consider it the best way to learn about determinants; but I do
+regard it as a road worth charting, and these notes are my attempt at doing so.
 \end{itemize}
 
 The notes include numerous exercises of varying difficulty, many of them
@@ -204,8 +206,8 @@ \section{\label{chp.intro}Introduction}
 theorem on their own instead of reading its proof.
 
 I have not meant these notes to be a textbook on any particular subject. For
-one thing, their does not map to any of the standard university courses, but
-rather straddles various subjects:
+one thing, their content does not map to any of the standard university
+courses, but rather straddles various subjects:
 
 \begin{itemize}
 \item Much of Chapter \ref{chp.binom} (on binomial coefficients) and Chapter
@@ -256,15 +258,15 @@ \subsection{Prerequisites}
 
 \begin{itemize}
 \item has a good grasp on basic school-level mathematics (integers, rational
-numbers, prime numbers, etc.);
+numbers, etc.);
 
 \item has some experience with proofs (mathematical induction, proof by
 contradiction, the concept of \textquotedblleft WLOG\textquotedblright, etc.)
 and mathematical notation (functions, subscripts, cases, what it means for an
 object to be \textquotedblleft well-defined\textquotedblright,
 etc.)\footnote{A great introduction into these matters (and many others!) is
 the free book \cite{LeLeMe16} by Lehman, Leighton and Meyer.
-(\textbf{Practical note:} As of 2017, this book is still undergoing frequent
+(\textbf{Practical note:} As of 2018, this book is still undergoing frequent
 revisions; thus, the version I am citing below might be outdated by the time
 you are reading this. I therefore suggest searching for possibly newer
 versions on the internet. Unfortunately, you will also find many older
@@ -324,14 +326,6 @@ \subsection{Prerequisites}
 
 [to add:
 
--- $\sum_{s\in S}$
-
--- $\prod_{s\in S}$
-
--- def $\sum_{i=a}^{b}$
-
--- def $\prod_{i=a}^{b}$
-
 -- explain empty sums \& products \& less-than-empty (do not \textquotedblleft
 sum backwards\textquotedblright)
 
@@ -94858,8 +94852,7 @@ \subsection{Solution to Exercise \ref{exe.block2x2.schur}}
 
 \subsection{Solution to Exercise \ref{exe.block2x2.jacobi}}
 
-Before we solve Exercise \ref{exe.block2x2.jacobi}, let us show two really
-simple facts:
+Before we solve Exercise \ref{exe.block2x2.jacobi}, let us show two really simple facts:
 
 \begin{lemma}
 \label{lem.sol.block2x2.jacobi.InIm}Let $n\in\mathbb{N}$ and $m\in\mathbb{N}$.
@@ -107344,9 +107337,9 @@ \subsection{\label{sect.sol.noncomm.polarization}Solution to Exercise
 \begin{array}
 [c]{c}%
 \text{here, we have renamed the}\\
-\text{summation index }i\text{ as }k
+\text{summation index }k\text{ as }i
 \end{array}
-\right) \nonumber\\
+\right)  \nonumber\\
 &  =\sum_{i\in\left[  n\right]  }\left[  i\in I\right]  v_{i}\nonumber\\
 &  =\underbrace{\sum_{\substack{i\in\left[  n\right]  ;\\i\in I}%
 }}_{\substack{=\sum_{i\in I}\\\text{(since }I\subseteq\left[  n\right]
@@ -107358,8 +107351,8 @@ \subsection{\label{sect.sol.noncomm.polarization}Solution to Exercise
 satisfies either }i\in I\text{ or }i\notin I\text{ (but not both)}\right)
 \nonumber\\
 &  =\sum_{i\in I}1v_{i}+\underbrace{\sum_{\substack{i\in\left[  n\right]
-;\\i\notin I}}0v_{i}}_{=0}=\sum_{i\in I}1v_{i}=\sum_{i\in I}v_{i}.
-\label{pf.lem.sol.noncomm.polarization.1.2}%
+;\\i\notin I}}0v_{i}}_{=0}=\sum_{i\in I}1v_{i}=\sum_{i\in I}v_{i}%
+.\label{pf.lem.sol.noncomm.polarization.1.2}%
 \end{align}
 
 
@@ -107377,11 +107370,11 @@ \subsection{\label{sect.sol.noncomm.polarization}Solution to Exercise
 \begin{align}
 \left(  \underbrace{\sum_{i\in I}v_{i}}_{\substack{=\sum_{k=1}^{n}\left[  k\in
 I\right]  v_{k}\\\text{(by (\ref{pf.lem.sol.noncomm.polarization.1.2}))}%
-}}\right)  ^{m}  &  =\left(  \sum_{k=1}^{n}\left[  k\in I\right]
-v_{k}\right)  ^{m}\nonumber\\
+}}\right)  ^{m} &  =\left(  \sum_{k=1}^{n}\left[  k\in I\right]  v_{k}\right)
+^{m}\nonumber\\
 &  =\sum_{\kappa:\left[  m\right]  \rightarrow\left[  n\right]  }\prod
 _{i=1}^{m}\left(  \left[  \kappa\left(  i\right)  \in I\right]  v_{\kappa
-\left(  i\right)  }\right) \nonumber\\
+\left(  i\right)  }\right)  \nonumber\\
 &  =\sum_{f:\left[  m\right]  \rightarrow\left[  n\right]  }\prod_{i=1}%
 ^{m}\left(  \left[  f\left(  i\right)  \in I\right]  v_{f\left(  i\right)
 }\right)  \label{pf.lem.sol.noncomm.polarization.1.4}%