(maxima.info)Introduction to linearalgebra
69.1 Introduction to linearalgebra
==================================
'linearalgebra' is a collection of functions for linear algebra.
Example:
(%i1) M : matrix ([1, 2], [1, 2]);
[ 1 2 ]
(%o1) [ ]
[ 1 2 ]
(%i2) nullspace (M);
[ 1 ]
[ ]
(%o2) span([ 1 ])
[ - - ]
[ 2 ]
(%i3) columnspace (M);
[ 1 ]
(%o3) span([ ])
[ 1 ]
(%i4) ptriangularize (M - z*ident(2), z);
[ 1 2 - z ]
(%o4) [ ]
[ 2 ]
[ 0 3 z - z ]
(%i5) M : matrix ([1, 2, 3], [4, 5, 6], [7, 8, 9]) - z*ident(3);
[ 1 - z 2 3 ]
[ ]
(%o5) [ 4 5 - z 6 ]
[ ]
[ 7 8 9 - z ]
(%i6) MM : ptriangularize (M, z);
[ 4 5 - z 6 ]
[ ]
[ 2 ]
[ 66 z 102 z 132 ]
[ 0 -- - -- + ----- + --- ]
(%o6) [ 49 7 49 49 ]
[ ]
[ 3 2 ]
[ 49 z 245 z 147 z ]
[ 0 0 ----- - ------ - ----- ]
[ 264 88 44 ]
(%i7) algebraic : true;
(%o7) true
(%i8) tellrat (MM [3, 3]);
3 2
(%o8) [z - 15 z - 18 z]
(%i9) MM : ratsimp (MM);
[ 4 5 - z 6 ]
[ ]
[ 2 ]
(%o9) [ 66 7 z - 102 z - 132 ]
[ 0 -- - ------------------ ]
[ 49 49 ]
[ ]
[ 0 0 0 ]
(%i10) nullspace (MM);
[ 1 ]
[ ]
[ 2 ]
[ z - 14 z - 16 ]
[ -------------- ]
(%o10) span([ 8 ])
[ ]
[ 2 ]
[ z - 18 z - 12 ]
[ - -------------- ]
[ 12 ]
(%i11) M : matrix ([1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12],
[13, 14, 15, 16]);
[ 1 2 3 4 ]
[ ]
[ 5 6 7 8 ]
(%o11) [ ]
[ 9 10 11 12 ]
[ ]
[ 13 14 15 16 ]
(%i12) columnspace (M);
[ 1 ] [ 2 ]
[ ] [ ]
[ 5 ] [ 6 ]
(%o12) span([ ], [ ])
[ 9 ] [ 10 ]
[ ] [ ]
[ 13 ] [ 14 ]
(%i13) apply ('orthogonal_complement, args (nullspace (transpose (M))));
[ 0 ] [ 1 ]
[ ] [ ]
[ 1 ] [ 0 ]
(%o13) span([ ], [ ])
[ 2 ] [ - 1 ]
[ ] [ ]
[ 3 ] [ - 2 ]
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