\documentclass[rep10,leqno]{report}
\voffset = -1in
\evensidemargin 0.1in
\oddsidemargin 0.1in
\textheight 9in
\textwidth 6in
\begin{document}
\normalsize
\baselineskip=12pt
\noindent
Input = 
$
\\-1\,
 c + (c\,
 a + d\,
 b)\,
 c\\
-1\,
 d + (c\,
 a + d\,
 b)\,
 d\\
c\,
 a + d\,
 b - a{}^{ T }\,
 c{}^{ T } - b{}^{ T }\,
 d{}^{ T }\\
-1\,
 c\,
 a - d\,
 b + a{}^{ T }\,
 c{}^{ T } + b{}^{ T }\,
 d{}^{ T }\\
a\,
 c - c{}^{ T }\,
 a{}^{ T }\\
-1\,
 a\,
 c + c{}^{ T }\,
 a{}^{ T }\\
a\,
 d - c{}^{ T }\,
 b{}^{ T }\\
-1\,
 a\,
 d + c{}^{ T }\,
 b{}^{ T }\\
b\,
 c - d{}^{ T }\,
 a{}^{ T }\\
-1\,
 b\,
 c + d{}^{ T }\,
 a{}^{ T }\\
b\,
 d - d{}^{ T }\,
 b{}^{ T }\\
-1\,
 b\,
 d + d{}^{ T }\,
 b{}^{ T }\\
(a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b] - \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\\
-1\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T } + Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\\
(a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T } - Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\\
-1\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b] + \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\\
-1\,
 a + a\,
 c\,
 a + a\,
 d\,
 b\\
-1\,
 b + b\,
 c\,
 a + b\,
 d\,
 b\\
-1\,
 a{}^{ T }\,
 a - b{}^{ T }\,
 b + (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\\
-1\,
 a{}^{ T }\,
 a - b{}^{ T }\,
 b + (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\\
Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b] - Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\\
a{}^{ T }\,
 c{}^{ T }\,
 a{}^{ T } + b{}^{ T }\,
 d{}^{ T }\,
 a{}^{ T } - a{}^{ T }\\
a{}^{ T }\,
 c{}^{ T }\,
 b{}^{ T } + b{}^{ T }\,
 d{}^{ T }\,
 b{}^{ T } - b{}^{ T }\\
c{}^{ T }\,
 (a{}^{ T }\,
 c{}^{ T } + b{}^{ T }\,
 d{}^{ T }) - c{}^{ T }\\
d{}^{ T }\,
 (a{}^{ T }\,
 c{}^{ T } + b{}^{ T }\,
 d{}^{ T }) - d{}^{ T }\\
\left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\,
 (a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T } - \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\\
Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b] - \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\\
-1\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b] + \left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\\
$
File Name = PseudoInverseOfColumn\\
NCMakeGB Iterations = 5\\
NCMakeGB on Digested Iterations = 0\\
SmallBasis Iterations = 0\\
SmallBasis on Knowns Iterations = 0\\
Deselect$\rightarrow \{\}$\\
UserSelect$\rightarrow \{\}$\\
UserUnknowns$\rightarrow \{\}$\\
NCShortFormulas$\rightarrow$-1\\
RR$\rightarrow $False\\
RRByCat$\rightarrow $False\\
SB$\rightarrow $False\\
SBByCat$\rightarrow $False\\
DegreeCap$\rightarrow $-1\\
DegreeSumCap$\rightarrow $-1\\
DegreeCapSB$\rightarrow $-1\\
DegreeSumCapSB$\rightarrow $-1\\
NCCV$\rightarrow $True\\
THE ORDER IS NOW THE FOLLOWING:\hfil\break
$
a<a{}^{ T }<b<b{}^{ T }<Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]<Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]<\left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }<\left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\ll
c<c{}^{ T }\ll
d<d{}^{ T }\\
$
\rule[2pt]{6in}{4pt}\hfil\break
\rule[2pt]{1.879in}{4pt}
\ YOUR SESSION HAS DIGESTED\ 
\rule[2pt]{1.879in}{4pt}\hfil\break
\rule[2pt]{1.923in}{4pt}
\ THE FOLLOWING RELATIONS\ 
\rule[2pt]{1.923in}{4pt}\hfil\break
\rule[2pt]{6in}{4pt}\hfil\break
THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR:\hfil\break
$\{c,
$ $
d,
$ $
c{}^{ T },
$ $
d{}^{ T },
$ $
\left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\}$
\smallskip\\
The corresponding rules are the following:\smallskip\\
\begin{minipage}{6in}
$
c\rightarrow Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 a{}^{ T }
$
\end{minipage}\medskip\\
\begin{minipage}{6in}
$
d\rightarrow Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 b{}^{ T }
$
\end{minipage}\medskip\\
\begin{minipage}{6in}
$
c{}^{ T }\rightarrow a\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]
$
\end{minipage}\medskip\\
\begin{minipage}{6in}
$
d{}^{ T }\rightarrow b\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]
$
\end{minipage}\medskip\\
\begin{minipage}{6in}
$
\left(Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\right){}^{ T }\rightarrow Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]
$
\end{minipage}\medskip\\
\rule[3pt]{6in}{.7pt}\\
The expressions with unknown variables $\{\}$\\
and knowns $\{a,
$ $
b,
$ $
Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b],
$ $
a{}^{ T },
$ $
b{}^{ T }\}$\smallskip\\
\begin{minipage}{6in}
$
Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 b{}^{ T }\,
 b\rightarrow -1\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 a{}^{ T }\,
 a + a{}^{ T }\,
 a\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b] + b{}^{ T }\,
 b\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
-1\,
 a\,
 (1 - b{}^{ T }\,
 b\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]) + a\,
 a{}^{ T }\,
 a\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]==0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
-1\,
 b\,
 (1 - a{}^{ T }\,
 a\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]) + b\,
 b{}^{ T }\,
 b\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]==0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
(a{}^{ T }\,
 a + b{}^{ T }\,
 b)\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b] - Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]==0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
-1\,
 (1 - b{}^{ T }\,
 b\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b])\,
 a{}^{ T } + a{}^{ T }\,
 a\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 a{}^{ T }==0
$
\end{minipage}\medskip \\
\begin{minipage}{6in}
$
-1\,
 (1 - a{}^{ T }\,
 a\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b])\,
 b{}^{ T } + b{}^{ T }\,
 b\,
 Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b]\,
 b{}^{ T }==0
$
\end{minipage}\\
\rule[2pt]{6in}{1pt}\hfil\break
\rule[2.5pt]{1.701in}{1pt}
\ USER CREATIONS APPEAR BELOW\ 
\rule[2.5pt]{1.701in}{1pt}\hfil\break
\rule[2pt]{6in}{1pt}\hfil\break
\rule[2pt]{6in}{4pt}\hfil\break
\rule[2pt]{1.45in}{4pt}
\ SOME RELATIONS WHICH APPEAR BELOW\ 
\rule[2pt]{1.45in}{4pt}\hfil\break
\rule[2pt]{2.18in}{4pt}
\ MAY BE UNDIGESTED\ 
\rule[2pt]{2.18in}{4pt}\hfil\break
\rule[2pt]{6in}{4pt}\hfil\break
THE FOLLOWING VARIABLES HAVE NOT BEEN SOLVED FOR:\hfil\break
$\{a,
$ $
b,
$ $
Pinv[a{}^{ T }\,
 a + b{}^{ T }\,
 b],
$ $
a{}^{ T },
$ $
b{}^{ T }\}$
\smallskip\\
\vspace{10pt}

\noindent
The time for preliminaries was 0:00:00\\
The time for NCMakeGB 1 was 0:00:00\\
The time for Remove Redundant 1 was 0:00:00\\
The time for the main NCMakeGB was 0:00:00\\
The time for Remove Redundant 2 was 0:00:00\\
The time for reducing unknowns was 0:00:00\\
The time for clean up basis was 0:00:00\\
The time for SmallBasis was 0:00:00\\
The time for CreateCategories was 0:00:00\\
The time for NCCV was 0:00:03\\
The time for RegularOutput was 0:00:00\\
The time for everything so far was 0:00:04\\
\end{document}
\end