deletions | additions       diff --git a/section_history.tex b/section_history.tex  index 7211b6e..8d0a6c5 100644  --- a/section_history.tex  +++ b/section_history.tex 
...
\paragraph{Bodewig (1956/9)~\cite{Bodewig1956}}  Vectors come first: first, but $n$-tuples don't actually play a part in the exposition.  Instead, they are glossed over immediately in favor of column vectors from the  ``matrices first'' school:  \begin{quote}  A vector $\mathbf v$ of order $n$ is an ordered set of $n$ numbers (so-called 
...
This book is also notable for its preface, explaining the author's preference a  systematic calculus of multiplication using basis vectors $\mathbf e_k$ to denote  operations like extracting the $k$th row of $\mathbf A$ as the product $\mathbf e_k A$.  p. 5 introduces basis vectors, $\mathbf e_k$, which he calls ``unity vectors',  and on p. 10 the vector of all ones, $\mathbf e$. With these vectors he points  out that you can write ``sum column'' and ``sum row'' vectors of $\mathbf M$. The  former he writes as:  \[  \mathbf{Me} = \mathbf M_{.1} + \mathbf M_{.2} + \dots + \mathbf M_{.n},  \]  being the column vector containg the sum over all the columns, and similarly for  the ``sum row'' vector $\mathbf{e^\prime M}$. In modern slicing notation we might  write \verb|M[:,1]| or even $M_{:,1}$ for $\mathbf M_{.1}$.  ... On p. 13 Bodewig states that $\mathbf e^\prime_i \mathbf A = \mathbf A_{i.}$  extracts the $i$th row of $\mathbf A$, and $\mathbf A \mathbf e_k = \mathbf A_{.k}$  extracts the $k$th column of $\mathbf A$, and furthermore that  $\mathbf e^\prime_i \mathbf A \mathbf e_k = a_{ij}$.  On p. 14 Bodewig points out that unit matrices $\mathbf E_{ik}$, which previously  appeared in abstract algebraic treatments, can be expressed ``as simple products  of two unit vectors'', i.e.\ $\mathbf E_{ik} = e^\prime_i e_k$.  Bodewig is very proud of his notation, saying on p. 13 that  ``[o]nly by using the formulae [above] will the calculation with matrices become a calculus.''  Bodewig presents bilinear and quadratic forms as $\mathbf{x^\prime A y}$ and  $\mathbf{x^\prime A x}$ respectively in chapter 5, p. 47.