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\newcommand{\RIX}{\end{array}\right]}
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\newcommand{\inner}[2]{\left\langle #1, #2 \right\rangle}
\newcommand{\GRP}{{\cal G}}        % Matrix Group
\newcommand{\HRP}{{\cal H}}        % Another Matrix Group
\newcommand{\GEN}{{\cal E}}        % Generators of matrix group
\newcommand{\VSP}{{\cal V}}        % Matrix Subspace
\newcommand{\WSP}{{\cal W}}        % Matrix Subspace
\newcommand{\USP}{{\cal U}}        % Matrix Subspace
\newcommand{\nullsp}{{\cal N}}     % Null space
\newcommand{\colsp}{{\cal R}}      % Column space
\newcommand{\eigsp}{{\cal E}}      % eigenspace
\newcommand{\geigsp}{{\cal G}}     % generalized eigenspace
\newcommand{\BMX}{\Phi}            % Basis for \VSP.
\newcommand{\AMX}{\Psi}            % Another such
\newcommand{\CMX}{\Upsilon}        % Another such
\newcommand{\bmx}{\nu}             % number of columns in \BMX
\newcommand{\amx}{\mu}             % number of columns in \AMX
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\begin{document}
\begin{center}\large \bf
  Matrix Algebra Using MINImal MATlab \\
\normalsize J Robbin \\
Errata
\end{center}

\bigskip

\ERR Exercise~57B. ``Then'' should be ``Prove that''.


\ERR Exercise~78C. The answer on page~475 has should read:
$$
   E_2E_3
          =  \MAT{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \RIX, \qquad
   (E_2E_3)^{-1} = E_3E_2
          =  \MAT{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \RIX.
$$

\ERR Page~79 ``\verb$#> A = [1 2; 3 4; 5 6]$''
   should be ``\verb$#> A = [1 2; 3 4; 4 5]$''

\ERR 84A Answer The last column of $R$ should be $2E_1-E_2$ not $E_1+3E_2$

\ERR Page~91. The listing of \verb$gj.m$ contains a bug.
The sixth line should be
\begin{quote}
       \verb$if r==m, free=[free, k:n]; return; end$
\end{quote}
not
\begin{quote}
       \verb$if r==m, free=[free, r+1:n]; return; end$
\end{quote}
This error may also appear in the disk  file \verb$gj.m$.
If so, fix it. Otherwise, some other \dotm files which depend on \verb$gj.m$
will function incorrectly.


\ERR Example~102A. Should read
$$
 H = \MAT{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \RIX,\quad
 T = \MAT{ccc} 0   & 0   & 1   \\ 1   & 0   & 0  \\ 0 & 1 & 0 \RIX, \quad
 H = \MAT{ccc} b_1 & c_1 & a_1 \\ b_2 & c_2 & a_2 \RIX.
$$

\ERR Exercise~103D. Answer on page~479:
 ``$Y=\MAT{r} -b \\ a \\ 0 \\ 0 \\ 0 \\ 0 \RIX$'' should be
 ``$Y=\MAT{r} -d \\ e \\ 0 \\ 0 \\ 0 \\ 0 \RIX$''

\ERR Proof of Lemma~106C ``$R=PD^{-1}$'' should be ``$R=DP^{-1}$''.

\ERR Exercise~122C the product $MY_3$ in the answer
on page~482 is wrong. The easiest fix is to change the problem
so $Y_3=\MAT{c} 8 \\ 0 \\ 0\RIX$. Also
``the is no'' on page ~482 should be ``there is no''.

\ERR Exercise~122D. Refers to Example~120A not~121B.

\ERR Exercise~122E. Answer on page~482 should read:
$Y_1=\MAT{r} 1\\1 \RIX$,
$Y_2=\MAT{r} 1\\2 \RIX$,
no.

\ERR Exercise~126A.  The second line should read
$$
 AX = x_1\col_1(A)+x_2\col_2(A)+\cdots+x_n\col_n(A)
$$

\ERR Exercise~129G
``$\colsp(B)\subset\nullsp(B)$''
should be ``$\colsp(B)\subset\colsp(A)$.''

\ERR Exercise~131A~(3) should be ``\verb$#> A=rand(3,5), B=rand(2,3)*A$

\ERR Theorem 132C.  ``for $j=1,2,\ldots,m$ in the statement of the
theorem and in the first line of the proof should be
``for $j=1,2,\ldots,n$. Also ``$A_j\in\VSP$'' in the proof should
be ``$A_j\in\WSP$.''

\ERR Corollary~144A. Last Line should read
$$
  \nullsp(\BMX) = \nullsp(P\BMX) = \nullsp(\BMX) = \{0\}.
$$

\ERR Example 147B ``$c_{11}=c_{12}=c_{22}=0$'' should be
``$c_{11}=c_{21}=c_{22}=0$''.


\ERR Exercise 148B Answer~(4) on page 487 should be
$(4)\quad\MAT{rrr}4 & -3 & -2 \\ -5 & 5 & 2\\ -2 & 2 & 1 \RIX$.

\ERR Exercise 148B answer on page 487: The first two columns of~(7)
form a basis for its range.

\ERR Exercise 148E ``$Y=\AMX X$'' should be ``$Y=AX$''.

\ERR Exercise 159A (b): ``defined by defined by'' should be ``defined by''
\ERR Page~180. Bottom line should read
$$
   Q_1 = \MAT{cc} A\AMX & \AMX_1\RIX, \qquad
   Q_2 = \MAT{cc} B\AMX & \AMX_2\RIX.
$$

\ERR Exercise~184F. Replace ``commute'' by ``are equal''.

\ERR Exercise 173B answer on page 489 ``$\nullsp(A)=\{0\}$''
should be``$\nullsp(\BMX)=\{0\}$''.

\ERR Exercise 165A answer. Replace \verb$Y$ by \verb$X$.

\ERR 176G Answer ``$\F^{m\times 1}\subset\nullsp(A)$'' should be
``$\F^{m\times 1}\subset\colsp(A)$''

\ERR 177B Answer In both parts~(1) and~(2) the sentence
 which begins ``If $A$ has a '' is not needed.

\ERR 177F Answer:  The question deals with right inverses not left invereses.
Step~(3) should read: If $R=MA$ has no row of zeros, we may find a right
inverse. $\tilde{\Omega}$ for $R$; then $\Omega=\tilde{\Omega}M$ is a right
inverse for $A=M^{-1}R$.

\ERR 200A Answer: The diagonal entry of the product should be
$\cos\theta\cos\phi-\sin\theta\sin\phi$ not
$\cos\theta\cos\phi-\sin\theta\cos\phi$.

\ERR Lemma 201A: $\BMX\in\F^{n\times\amx}$ should be $\BMX\in\F^{n\times k}$. 

\ERR Page 207 ``$z_{kj}=\inner{V_k}{U_j}$'' should be
              ``$z_{jk}=\inner{V_k}{U_j}$''.

\ERR Exercise~260A. In the answer on page 500 \verb$solve$
should be \verb$randsoln$.

\ERR The answer to Exercise~264B on page~501 contains an
obvious reversal error.


\end{document}

% \ERR Page~64. The restriction
% ``Type \verb$D=diag([1 2 3 4])$ not \verb$D=diag(1 2 3 4)$''
% does not apply to later versions of MINIMAT or to Matlab -
% either form works.
% Also there should be an index entry for \verb$diag$ pointing to
% this page  and \verb$diag$ should appear in the table on page~450.
