Exact Linear Algebra for SAGE - William Stein - University of ...
Exact Linear Algebra for SAGE - William Stein - University of ...
Exact Linear Algebra for SAGE - William Stein - University of ...
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Echelon Forms <strong>of</strong> Matrices<br />
Computing Echelon Forms<br />
Decomposing Spaces Under the Action <strong>of</strong> Matrix<br />
Echelon <strong>for</strong>m example 2<br />
Example<br />
Notice that the entries <strong>of</strong> the reduced row echelon <strong>for</strong>m can be rationals with<br />
large denominators even though the entries <strong>of</strong> the original matrix A are<br />
integers. Another example is the simple looking matrix<br />
0<br />
B<br />
@<br />
whose echelon <strong>for</strong>m is<br />
0<br />
1 0 0 0<br />
42<br />
B<br />
@<br />
0<br />
0<br />
1<br />
0<br />
0<br />
1<br />
0<br />
0 − 83<br />
0 0 0 1<br />
−9 6 7 3 1 0 0 0<br />
−10 3 8 2 0 1 0 0<br />
3 −6 2 8 0 0 1 0<br />
−8 −6 −8 6 0 0 0 1<br />
1025<br />
716<br />
3075<br />
1025<br />
184<br />
1025<br />
92<br />
− 1025<br />
641<br />
− 3075<br />
133<br />
1025<br />
159<br />
− 1025<br />
1<br />
25<br />
2<br />
− 75<br />
1<br />
25<br />
2<br />
25<br />
1<br />
C<br />
A<br />
9<br />
− 205<br />
7<br />
− 615<br />
23<br />
− 410<br />
9<br />
410<br />
<strong>William</strong> <strong>Stein</strong> <strong>Exact</strong> <strong>Linear</strong> <strong>Algebra</strong> <strong>for</strong> <strong>SAGE</strong><br />
1<br />
C<br />
A .