Evaluating if two basis sets span the same subspace in maxima

Question

Is there some function in maxima for evaluating if two bases span the same subspace.

For example, I ask my student to calculate a basis of the null space of a given matrix. In my solution, I have a specific basis. But I want to make sure that if the student uses a different but correct calculation for determining the basis, that the answer will still be graded as correct.

(%i1) sol: nullspace(matrix([1,0,0],[0,0,0],[0,0,0]));
                                   [ 0 ]  [ 0 ]
                                   [   ]  [   ]
(%o1)                         span([ 0 ], [ 1 ])
                                   [   ]  [   ]
                                   [ 1 ]  [ 0 ]
(%i2) answer_student: span(matrix([0],[0],[2]),matrix([0],[2],[0]));
                                   [ 0 ]  [ 0 ]
                                   [   ]  [   ]
(%o2)                         span([ 0 ], [ 2 ])
                                   [   ]  [   ]
                                   [ 2 ]  [ 0 ]
(%i3) is(sol = answer_student);
(%o3)                                false

Answer 1

The nullspace function doesn't appear to have a provision for normalizing the vectors; as the documentation says, it returns "a" set of basis vectors. But I guess you can normalize them yourself easily enough. E.g.

f : lambda ([v], v/sqrt(v.v));
map (f, whatever_output_of_nullspace);

Note that v/sqrt(v.v) normalizes the column v assuming default values for some of the matrix operation flags.

Answer 2

I don't think there's a built-in way to do this. You can reduce your problem to finding dimensions and finding sums of subspaces, since a subspace U equals another subspace V if and only if dim U = dim V = dim (U + V).

We can represent subspaces in the usual way: matrices whose columns form a spanning set for the subspace.

Here's a function which takes a matrix a and a list of vectors l of the same height as a and returns whether or not l spans the nullspace of a:

spans_nullspace(a, l) := block(
 ker_a : apply(addcol, args(nullspace(a))),
 V : apply(addcol,l),
 ker_a_plus_V : mat_unblocker(matrix([ker_a,V])),
 is(nullity(a) = rank(ker_a_plus_V) and rank(ker_a) = nullity(a))
 );

Some explanation:

• ker_a is a matrix whose columns are the basis for the nullspace of a found by nullspace(a).
• V : apply(addcol,l) is the matrix whose columns are the vectors in l, so it represents the student's answer.
• mat_unblocker(matrix([ker_a,V])) is the matrix obtained by putting ker_a next to V, so it represents the sum of the kernel of a and the student's subspace.

Another way to check equality of subspaces in this representation is to use the fact that two same-size matrices have the same row space if and only if their row reduced echelon forms are equal. Since we care about column space, we can transpose, pad the matrices so they're the same size, then check for equality of the RREFs. But maxima doesn't have a RREF function built in (it does have echelon form) so you still have to write some code.