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This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) (preloaded format=latex)
entering extended mode
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LaTeX2e <2014/05/01>
Babel <3.9l> and hyphenation patterns for 2 languages loaded.
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Document Class: article 2014/09/29 v1.4h Standard LaTeX document class
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l.44 \begin{document}
                     

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l.45 \begin{multicols*}
                       {3}
! Undefined control sequence.
<recently read> \url 
                     
l.50 \url
         {http://wiki.sagemath.org/quickref}\\
! Undefined control sequence.
\warn ->\bf \color 
                   {red}
l.58 {\warn
            Caution}: First entry of a vector is numbered 0\\
! Undefined control sequence.
\ex ->\color 
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l.59 {\ex
         \verb!u = vector(QQ, [1, 3/2, -1])!} length 3 over rationals\\
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\ex ->\color 
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l.60 {\ex
         \verb!v = vector(QQ, {2:4, 95:4, 210:0})!}\\
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\ex ->\color 
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l.65 {\ex
         \verb!u = vector(QQ, [1, 3/2, -1])!}\\
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l.66 {\ex
         \verb!v = vector(ZZ, [1, 8, -2])!}\\
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\ex ->\color 
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l.67 {\ex
         \verb!2*u - 3*v!} linear combination\\
! Undefined control sequence.
\ex ->\color 
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l.68 {\ex
         \verb!u.dot_product(v)!}\\
! Undefined control sequence.
\ex ->\color 
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l.69 {\ex
         \verb!u.cross_product(v)!}  order: \verb!u!$\times$\verb!v!\\
! Undefined control sequence.
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l.70 {\ex
         \verb!u.inner_product(v)!}  inner product matrix from parent\\
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l.71 {\ex
         \verb!u.pairwise_product(v)!} vector as a result\\
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l.72 {\ex
         \verb!u.norm() == u.norm(2)!} Euclidean norm\\
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l.73 {\ex
         \verb!u.norm(1)!} sum of entries\\
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l.74 {\ex
         \verb!u.norm(Infinity)!} maximum entry\\
! Undefined control sequence.
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l.75 {\ex
         \verb!A.gram_schmidt()!} converts the rows of matrix \verb!A!
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l.79 {\warn
            Caution}: Row, column numbering begins at 0\\
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\ex ->\color 
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l.80 {\ex
         \verb!A = matrix(ZZ, [[1,2],[3,4],[5,6]])!}\\
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l.82 {\ex
         \verb!B = matrix(QQ, 2, [1,2,3,4,5,6])!}\\
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l.84 {\ex
         \verb!C = matrix(CDF, 2, 2, [[5*I, 4*I], [I, 6]])!}\\
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l.86 {\ex
         \verb!Z = matrix(QQ, 2, 2, 0)!} zero matrix\\
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l.87 {\ex
         \verb!D = matrix(QQ, 2, 2, 8)!}\\
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l.89 {\ex
         \verb!E = block_matrix([[P,0],[1,R]])!}, very flexible input\\
! Undefined control sequence.
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l.90 {\ex
         \verb!II = identity_matrix(5)!} $5\times5$ identity matrix\\
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l.92 {\ex
         \verb!J = jordan_block(-2,3)!}\\
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l.94 {\ex
         \verb!var('x y z'); K = matrix(SR, [[x,y+z],[0,x^2*z]])!}\\
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l.96 {\ex
         \verb!L = matrix(ZZ, 20, 80, {(5,9):30, (15,77):-6})!}\\

Overfull \hbox (201.08641pt too wide) in paragraph at lines 79--98
| $\OT1/cmr/m/n/12 3 \OMS/cmsy/m/n/12 ^^B \OT1/cmr/m/n/12 3$ ma-trix, $\OMS/cms
y/m/n/12 ^^@\OT1/cmr/m/n/12 2$ on di-ag-o-nal, $1$'s on super-diagonal blue[]\O
T1/cmtt/m/n/12 var('x y z'); K = matrix(SR, [[x,y+z],[0,x^2*z]]) 
[1]
! Undefined control sequence.
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l.101 {\ex
          \verb!u = vector(QQ, [1,2,3])!},\quad{\ex\verb!v = vector(QQ, [1,2...

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l.101 ...\verb!u = vector(QQ, [1,2,3])!},\quad{\ex
                                                  \verb!v = vector(QQ, [1,2]...

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l.102 {\ex
          \verb!A = matrix(QQ, [[1,2,3],[4,5,6]])!}\\
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l.103 {\ex
          \verb!B = matrix(QQ, [[1,2],[3,4]])!}\\
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l.104 {\ex
          \verb!u*A!},\quad
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l.105 {\ex
          \verb!A*v!},\quad
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l.106 {\ex
          \verb!B*A!},\quad
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l.107 {\ex
          \verb!B^6!},\quad
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l.108 {\ex
          \verb!B^(-3)!} all possible\\
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l.109 {\ex
          \verb!B.iterates(v, 6)!}  produces $vB^0, vB^1,\dots, vB^5$\\
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l.111 {\ex
          \verb!f(x)=x^2+5*x+3!} then {\ex\verb!f(B)!} is possible\\
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l.111 {\ex\verb!f(x)=x^2+5*x+3!} then {\ex
                                          \verb!f(B)!} is possible\\
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l.112 {\ex
          \verb!B.exp()!} matrix exponential, i.e.\ $\sum_{k=0}^{\infty}\fra...

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l.116 {\ex
          \verb!M = MatrixSpace(QQ, 3, 4)!} is space of $3\times 4$ matrices\\
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l.117 {\ex
          \verb!A = M([1,2,3,4,5,6,7,8,9,10,11,12])!}\\
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          \verb!M.basis()!}\\
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          \verb!M.dimension()!}\\
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          \verb!M.zero_matrix()!}
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l.125 {\ex
          \verb!5*A+2*B!} linear combination\\
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l.126 {\ex
          \verb!A.inverse()!}, {\ex\verb!A^(-1)!}, {\ex\verb!~A!}, singular ...

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l.126 {\ex\verb!A.inverse()!}, {\ex
                                   \verb!A^(-1)!}, {\ex\verb!~A!}, singular ...

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l.126 ...b!A.inverse()!}, {\ex\verb!A^(-1)!}, {\ex
                                                  \verb!~A!}, singular is \v...

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l.127 {\ex
          \verb!A.transpose()!}\\
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l.128 {\ex
          \verb!A.conjugate()!} entry-by-entry complex conjugates\\
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l.129 {\ex
          \verb!A.conjugate_transpose()!}\\
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l.130 {\ex
          \verb!A.antitranspose()!} transpose + reverse orderings\\
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l.131 {\ex
          \verb!A.adjoint()!}  matrix of cofactors\\
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l.132 {\ex
          \verb!A.restrict(V)!} restriction to invariant subspace \verb!V!
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l.137 {\warn
             Caution}: first row is numbered 0\\
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l.138 {\ex
          \verb!A.rescale_row(i,a)!} \verb!a*!(row \verb!i!)\\
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l.139 {\ex
          \verb!A.add_multiple_of_row(i,j,a)!} \verb!a*!(row \verb!j!) + row...

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l.140 {\ex
          \verb!A.swap_rows(i,j)!}\\
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l.142 For a new matrix, use e.g.\ {\ex
                                      \verb!B = A.with_rescaled_row(i,a)!}
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l.146 {\ex
          \verb!A.rref()!},
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l.147 {\ex
          \verb!A.echelon_form()!},
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l.148 {\ex
          \verb!A.echelonize()!}\\
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l.149 {\warn
             Note}: \verb!rref()! promotes matrix to fraction field\\
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l.150 {\ex
          \verb!A = matrix(ZZ,[[4,2,1],[6,3,2]])!}\\
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l.152 {\ex
          \verb!A.rref()!} & {\ex\verb!A.echelon_form()!}\\
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l.152 {\ex\verb!A.rref()!} & {\ex
                                 \verb!A.echelon_form()!}\\
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l.163 {\ex
          \verb!A.pivots()!} indices of columns spanning column space\\
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l.164 {\ex
          \verb!A.pivot_rows()!} indices of rows spanning row space
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l.168 {\warn
             Caution}: row, column numbering begins at 0\\
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l.169 {\ex
          \verb!A.nrows()!}, {\ex\verb!A.ncols()!}\\
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l.169 {\ex\verb!A.nrows()!}, {\ex
                                 \verb!A.ncols()!}\\
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l.170 {\ex
          \verb!A[i,j]!} entry in row \verb!i! and column \verb!j!\\
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l.171 {\ex
          \verb!A[i]!} row \verb!i! as immutable Python tuple.  Thus,\\
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l.172 \skipin{\warn
                    Caution}: OK: \verb!A[2,3] = 8!, Error: \verb!A[2][3] = ...

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l.173 {\ex
          \verb!A.row(i)!} returns row \verb!i! as Sage vector\\
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l.174 {\ex
          \verb!A.column(j)!} returns column \verb!j! as Sage vector\\
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l.175 {\ex
          \verb!A.list()!} returns single Python list, row-major order\\
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l.176 {\ex
          \verb!A.matrix_from_columns([8,2,8])!}\\
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l.178 {\ex
          \verb!A.matrix_from_rows([2,5,1])!}\\
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l.180 {\ex
          \verb!A.matrix_from_rows_and_columns([2,4,2],[3,1])!}\\
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l.182 {\ex
          \verb!A.rows()!}  all rows as a list of tuples\\
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l.183 {\ex
          \verb!A.columns()!}  all columns as a list of tuples\\
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l.184 {\ex
          \verb!A.submatrix(i,j,nr,nc)!}\\
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l.186 {\ex
          \verb!A[2:4,1:7]!}, {\ex\verb!A[0:8:2,3::-1]!} Python-style list s...

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l.186 {\ex\verb!A[2:4,1:7]!}, {\ex
                                  \verb!A[0:8:2,3::-1]!} Python-style list s...

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l.190 {\ex
          \verb!A.augment(B)!}  \verb!A! in first columns, matrix \verb!B! t...

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l.191 {\ex
          \verb!A.stack(B)!}  \verb!A! in top rows, \verb!B! below; \verb!B!...

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l.192 {\ex
          \verb!A.block_sum(B)!}  Diagonal, \verb!A! upper left, \verb!B! lo...

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l.193 {\ex
          \verb!A.tensor_product(B)!}  Multiples of \verb!B!, arranged as in...

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l.197 {\ex
          \verb!A.rank()!},\quad{\ex\verb!A.right_nullity()!}\\
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l.197 {\ex\verb!A.rank()!},\quad{\ex
                                    \verb!A.right_nullity()!}\\
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l.198 {\ex
          \verb!A.left_nullity() == A.nullity()!}\\
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l.199 {\ex
          \verb!A.determinant() == A.det()!}\\
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l.200 {\ex
          \verb!A.permanent()!},\quad{\ex\verb!A.trace()!}\\
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l.200 {\ex\verb!A.permanent()!},\quad{\ex
                                         \verb!A.trace()!}\\
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l.201 {\ex
          \verb!A.norm() == A.norm(2)!} Euclidean norm\\
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l.202 {\ex
          \verb!A.norm(1)!} largest column sum\\
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l.203 {\ex
          \verb!A.norm(Infinity)!} largest row sum\\
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l.204 {\ex
          \verb!A.norm('frob')!} Frobenius norm
[3]
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l.210 {\ex
          \verb!.is_zero()!};
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l.211 {\ex
          \verb!.is_symmetric()!};
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l.212 {\ex
          \verb!.is_hermitian()!};\\
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l.214 {\ex
          \verb!.is_square()!};
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l.215 {\ex
          \verb!.is_orthogonal()!};
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l.216 {\ex
          \verb!.is_unitary()!};\\
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l.218 {\ex
          \verb!.is_scalar()!};
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l.219 {\ex
          \verb!.is_singular()!};
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l.220 {\ex
          \verb!.is_invertible()!};\\
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l.222 {\ex
          \verb!.is_one()!};
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l.223 {\ex
          \verb!.is_nilpotent()!};
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l.224 {\ex
          \verb!.is_diagonalizable()!}
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\warn ->\bf \color 
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l.231 {\warn
             Note}: Contrast behavior for exact rings (\verb!QQ!) vs.\ \verb...

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l.232 {\ex
          \verb!A.charpoly('t')!}  no variable specified defaults to \verb!x!\\

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l.233 {\ex
          \verb!  A.characteristic_polynomial() == A.charpoly()!}\\
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l.234 {\ex
          \verb!A.fcp('t')!}  factored characteristic polynomial\\
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l.235 {\ex
          \verb!A.minpoly()!}  the minimum polynomial\\
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l.236 {\ex
          \verb!  A.minimal_polynomial() == A.minpoly()!}\\
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l.237 {\ex
          \verb!A.eigenvalues()!} unsorted list, with mutiplicities\\
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l.238 {\ex
          \verb!A.eigenvectors_left()!} vectors on left, \verb!_right! too\\
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l.243 {\ex
          \verb!A.eigenmatrix_right()!} vectors on right, \verb!_left! too\\
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l.252 {\warn
             Note}: availability depends on base ring of matrix,\\
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l.255 {\ex
          \verb!A.jordan_form(transformation=True)!}\\
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l.260 {\ex
          \verb!A.smith_form()!} triple with: \verb!D == U*A*V!\\
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l.264 {\ex
          \verb!A.LU()!} triple with: \verb!P*A == L*U!\\
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l.268 {\ex
          \verb!A.QR()!} pair with: \verb!A == Q*R!\\
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l.271 {\ex
          \verb!A.SVD()!} triple with: \verb!A == U*S*(V-conj-transpose)!\\
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l.276 {\ex
          \verb!A.schur()!} pair with: \verb!A == Q*T*(Q-conj-transpose)!\\
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l.280 {\ex
          \verb!A.rational_form()!}, aka Frobenius form\\
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l.282 {\ex
          \verb!A.symplectic_form()!}\\
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l.283 {\ex
          \verb!A.hessenberg_form()!}\\
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l.284 {\ex
          \verb!A.cholesky()!} (needs work)
[4]
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l.288 {\ex
          \verb!A.solve_right(B)!} \verb!_left! too\\
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l.290 {\ex
          \verb!A = matrix(QQ, [[1,2],[3,4]])!}\\
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l.291 {\ex
          \verb!b = vector(QQ, [3,4])!}, then {\ex\verb!A\b!} is solution {\...

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l.291 {\ex\verb!b = vector(QQ, [3,4])!}, then {\ex
                                                  \verb!A\b!} is solution {\...

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\ex ->\color 
             {blue}
l.291 ...!}, then {\ex\verb!A\b!} is solution {\ex
                                                  \verb!(-2, 5/2)!}\par
! Undefined control sequence.
\ex ->\color 
             {blue}
l.294 {\ex
          \verb!VectorSpace(QQ, 4)!}  dimension 4, rationals as field\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.295 {\ex
          \verb!VectorSpace(RR, 4)!}  ``field'' is 53-bit precision reals\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.296 {\ex
          \verb!VectorSpace(RealField(200), 4)!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.298 {\ex
          \verb!CC^4!} 4-dimensional, 53-bit precision complexes\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.299 {\ex
          \verb!Y = VectorSpace(GF(7), 4)!}  finite\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.300 \skipin{\ex
                 \verb!Y.list()!} has $7^4=2401$ vectors
! Undefined control sequence.
\ex ->\color 
             {blue}
l.304 {\ex
          \verb!V.dimension()!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.305 {\ex
          \verb!V.basis()!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.306 {\ex
          \verb!V.echelonized_basis()!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.307 {\ex
          \verb!V.has_user_basis()!} with non-canonical basis?\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.308 {\ex
          \verb!V.is_subspace(W)!} \verb!True! if \verb!W! is a subspace of ...

! Undefined control sequence.
\ex ->\color 
             {blue}
l.309 {\ex
          \verb!V.is_full()!} rank equals degree (as module)?\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.310 {\ex
          \verb!Y = GF(7)^4!},\quad{\ex\verb!T = Y.subspaces(2)!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.310 {\ex\verb!Y = GF(7)^4!},\quad{\ex
                                       \verb!T = Y.subspaces(2)!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.317 {\ex
          \verb!span([v1,v2,v3], QQ)!} span of list of vectors over ring
! Undefined control sequence.
\ex ->\color 
             {blue}
l.322 {\ex
          \verb!A.left_kernel() == A.kernel()!} \verb!right_! too\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.323 {\ex
          \verb!A.row_space() == A.row_module()!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.324 {\ex
          \verb!A.column_space() == A.column_module()!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.325 {\ex
          \verb!A.eigenspaces_right()!} vectors on right, \verb!_left! too\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.327 {\ex
          \verb!A.eigenspaces_right(format='galois')!}\\
[5]
! Undefined control sequence.
\ex ->\color 
             {blue}
l.331 {\ex
          \verb!V.quotient(W)!} quotient of \verb!V! by subspace \verb!W!\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.332 {\ex
          \verb!V.intersection(W)!} intersection of \verb!V! and \verb!W!\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.333 {\ex
          \verb!V.direct_sum(W)!} direct sum of \verb!V! and \verb!W!\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.334 {\ex
          \verb!V.subspace([v1,v2,v3])!} specify basis vectors in a list
! Undefined control sequence.
\warn ->\bf \color 
                   {red}
l.338 {\warn
             Note:} Algorithms may depend on representation\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.342 {\ex
          \verb!.is_dense()!}, {\ex\verb!.is_sparse()!}  to check\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.342 {\ex\verb!.is_dense()!}, {\ex
                                   \verb!.is_sparse()!}  to check\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.343 {\ex
          \verb!A.sparse_matrix()!} returns sparse version of \verb!A!\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.344 {\ex
          \verb!A.dense_rows()!} returns dense row vectors of \verb!A!\\
! Undefined control sequence.
\warn ->\bf \color 
                   {red}
l.349 {\warn
             Note:} Many algorithms depend on the base ring\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.350 {\ex
          \verb!<object>.base_ring(R)!} for vectors, matrices,\dots\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.352 {\ex
          \verb!<object>.change_ring(R)!} for vectors, matrices,\dots\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.354 {\ex
          \verb!R.is_ring()!},\quad{\ex\verb!R.is_field()!},\quad{\ex\verb!R...

! Undefined control sequence.
\ex ->\color 
             {blue}
l.354 {\ex\verb!R.is_ring()!},\quad{\ex
                                       \verb!R.is_field()!},\quad{\ex\verb!R...

! Undefined control sequence.
\ex ->\color 
             {blue}
l.354 ...},\quad{\ex\verb!R.is_field()!},\quad{\ex
                                                  \verb!R.is_exact()!}\\[2pt]
! Undefined control sequence.
\ex ->\color 
             {blue}
l.356 \skipin{\ex
                 \verb!ZZ!}\quad integers, ring\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.357 \skipin{\ex
                 \verb!QQ!}\quad rationals, field\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.358 \skipin{\ex
                 \verb!AA!, \verb!QQbar!}\quad algebraic number fields, exact\\

! Undefined control sequence.
\ex ->\color 
             {blue}
l.359 \skipin{\ex
                 \verb!RDF!}\quad real double field, inexact\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.360 \skipin{\ex
                 \verb!CDF!}\quad complex double field, inexact\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.361 \skipin{\ex
                 \verb!RR!}\quad 53-bit reals, inexact, not same as {\ex\ver...

! Undefined control sequence.
\ex ->\color 
             {blue}
l.361 ...d 53-bit reals, inexact, not same as {\ex
                                                  \verb!RDF!}\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.362 \skipin{\ex
                 \verb!RealField(400)!}\quad 400-bit reals, inexact\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.363 \skipin{\ex
                 \verb!CC!,\quad\verb!ComplexField(400)!}\quad complexes, too\\

! Undefined control sequence.
\ex ->\color 
             {blue}
l.364 \skipin{\ex
                 \verb!RIF!}\quad real interval field\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.365 \skipin{\ex
                 \verb!GF(2)!}\quad mod 2, field, specialized implementations\\

! Undefined control sequence.
\ex ->\color 
             {blue}
l.366 \skipin{\ex
                 \verb!GF(p) == FiniteField(p)!}\quad \verb!p! prime, field\\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.367 \skipin{\ex
                 \verb!Integers(6)!}\quad integers mod 6, ring only \\
! Undefined control sequence.
\ex ->\color 
             {blue}
l.368 \skipin{\ex
                 \verb!CyclotomicField(7)!}\quad rationals with 7$^{\rm th}$...

! Undefined control sequence.
\ex ->\color 
             {blue}
l.369 \skipin{\ex
                 \verb!QuadraticField(-5, 'x')!}\quad rationals with \verb!x...

! Undefined control sequence.
\ex ->\color 
             {blue}
l.370 \skipin{\ex
                 \verb!SR!}\quad ring of symbolic expressions
[6]

! LaTeX Error: \begin{document} on input line 44 ended by \end{multicols*}.

See the LaTeX manual or LaTeX Companion for explanation.
Type  H <return>  for immediate help.
 ...                                              
                                                  
l.385 \end{multicols*}
                      
[7] (./latex_1c6e71d75ec9edc1d3e893ab4c724d503905c830_p.aux) )
(\end occurred inside a group at level 1)

### semi simple group (level 1) entered at line 44 (\begingroup)
### bottom level
(see the transcript file for additional information)
Output written on latex_1c6e71d75ec9edc1d3e893ab4c724d503905c830_p.dvi (7 pages
, 15808 bytes).
Transcript written on latex_1c6e71d75ec9edc1d3e893ab4c724d503905c830_p.log.

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