% !TeX program = lualatex
% =====================================================================
%  algebra.tex
%  Linear algebra, objects then vocabulary. Objects: a matrix, a
%  determinant and a square-bracket matrix, the augmentation bar of a
%  linear system, and the system block that aligns on the first relation.
%  Vocabulary: the named operators kernel, image, rank, span, trace,
%  transpose, inverse, comatrix, spectrum and adjoint. One line is one
%  row throughout.
% =====================================================================
\documentclass[
  margins=18,
  font=Latin Modern Roman,
  size=12,
  linespread=1.4,
  lang=en
]{scholatex}
\begin{document}

let title = <Red b 18pt c>
let h1    = <Navy b section>
let raw   = <Gray i>

<title>scholatex — algebra

% =====================================================================
<h1>Matrix blocks
% =====================================================================

A matrix is a block: one line is one row, ; separates the entries.
Every cell goes through the mini-language, so $2x + 1$ or $1/2$ work
inside a cell.

<matrix>{
	1 ; 2 ; 3
	4 ; 5 ; 6
}

<raw>The raw form needs a pmatrix environment, an ampersand between
every entry and a double backslash at every row end.

The same block under two other names draws a determinant or square
brackets:

<det>{
	a ; b
	c ; d
}

<bmatrix>{
	2x + 1 ; 0
	0 ; 1/2
}

% =====================================================================
<h1>Augmented matrix
% =====================================================================

A single | inside a row draws the augmentation bar at that column on
every row — how a linear system is set up for elimination:

<bmatrix>{
	2 ; 1 | 7
	1 ; -1 | 1
}

The bar is allowed on <<matrix>> and <<bmatrix>>, never on
<<det>>. A bar at a row edge, or misaligned across rows, raises a
clear scholatex: error naming the line.

% =====================================================================
<nextpage h1>Systems of equations
% =====================================================================

A system aligns automatically on the first relational operator, so
equalities and inequalities mix freely. One equation per line, no
separator:

<system>{
	2x + 3y = 7
	x - y = 1
}

<system>{
	2x + 3 <= 7
	x >= 0
}

A solution set then states itself in the language of sets — a pair in
$RR^2$ subject to its constraints:

<box line:Navy fill:AliceBlue radius:3 title:{The solution set}>{
	$S = {{ (x, y) in RR^2 : 2x + 3y = 7 and x - y = 1 }}$
}

% =====================================================================
<nextpage h1>Kernel, image, rank
% =====================================================================

Beyond the matrix objects, the vocabulary of linear maps reads as named
operators. $ker(f)$ is the kernel, $im(f)$ the image, $rank(A)$ the rank;
they tie together in the rank–nullity theorem.

<box line:Teal fill:MintCream radius:3 title:{Rank–nullity}>{
	$dim(ker(f)) + rank(f) = dim(E)$
}

% =====================================================================
<h1>Span, trace, determinant
% =====================================================================

A family generates a subspace with $span(u, v, w)$; a square matrix carries
a trace $tr(A)$ and a determinant $det(A)$.

<box line:Teal fill:MintCream radius:3 title:{Trace and determinant of a product}>{
	$tr(A B) = tr(B A)$

	$det(A B) = det(A) det(B)$
}

% =====================================================================
<h1>Transpose, inverse, comatrix
% =====================================================================

The transpose is $transpose(A)$, which sets the upright transpose mark; the
inverse is $inv(A)$, the same as writing $A^(-1)$. The comatrix $com(A)$
enters the inversion formula.

<box line:Teal fill:MintCream radius:3 title:{Inversion by the comatrix}>{
	$inv(A) = transpose(com(A)) / det(A)$
}

% =====================================================================
<h1>Spectrum and adjoint
% =====================================================================

The set of eigenvalues is the spectrum $eigen(A)$, and the adjoint is
$adj(A)$. A matrix is diagonalisable when a basis of eigenvectors spans the
whole space.

<box line:Teal fill:MintCream radius:3 title:{Characteristic property}>{
	$lambda in eigen(A) <=> det(A - lambda I) = 0$
}

\end{document}