Dear Francois, * I'd like to concentrate on your function "columnBasis". I think that the other functions are less important.
+ First of all, I'd *really* like to call it columnnSpace: M -> List Col. This is standard terminology, as far as I can tell. At least, it's also the term used on wikipedia. As a synonym, we probably should have - analogous to basis$ALGPKG, a function basis: List Col -> List Col. But maybe this should go into a different package, maybe VectorCollectionFunctions(R: Ring, L: Collection Vector R) or some such. But really, that should be done in a second step. + The documentation could link to http://en.wikipedia.org/wiki/Row_space + Your code can be made slightly more efficient. + We only need an Euclidean domain for your code (and the maths) to work. In fact, Integers are a prime example. Although, I must admit that in this case the names are slightly confusing, because we obtain a Q-basis of the columnSpace, where Q is the quotient field of R. The code should actually go into MATCAT$matcat.spad.pamphlet. You also have to make it available for RectangularMatrix. There should be an entry in RectangularMatrixCategory$matcat.spad.pamphlet, right after rowEchelon. The implementation can go as default into this category, because you need only row and rowEchelon. * if you like to invest more time, I think it would be *very* worthwhile to check whether we can make RectangularMatrixCategory inherit from MatrixCategory. As far as I can see, the only problem is that currently a domain having DirectProductCategory(n, R) is not a FiniteLinearAggregate R. But otherwise, I do not see a good reason for that restriction. In any case, that's a DIFFERENT task. ------------------------------------------------------------------------------- \verb|columnSpace| extracts from the columns of $M$ a basis for the space they generate. The columns of $M$ that generate the space correspond to those columns in the row echelon form of $M$ whose first non-zero entry is non-zero. As an example, let $M$ be the matrix \begin{verbatim} +2 1 0 2+ | | |2 0 3 0| | | +2 1 0 1+ \end{verbatim} Its row-echelon form is \begin{verbatim} +2 0 3 0+ | | |0 1 - 3 0| | | +0 0 0 1+ \end{verbatim} And indeed, \verb|[[2,2,2],[1,0,1],[2,0,1]]| is a $\mathbb Q$-basis of the column space. columnSpace M == M2 := rowEchelon M basis: List Col := [] n: Integer := ncols M m: Integer := nrows M indRow: Integer := 1 for k in 1..n while indRow <= m repeat if not zero?(M2.(indRow, k)) then basis := cons(column(M, k), basis) indRow := indRow + 1 reverse! basis Reversing the extracted columns is not strictly necessary, but makes the result look more natural. ------------------------------------------------------------------------------- + concerning the other functions, I think that the only one worth adding is the one about the intersection of two bases, which I'd like to call something like "intersection", and which could also live in VectorCollectionFunctions. Again that's ANOTHER task. * Finally, the above *really* suggests that we might want to have a domain whose elements are bases of a given vector space. But I guess that we would need aldor for that to do it properly. (It's similar to the problem that an element of PermutationGroup is not a Group...) Hm, but the domain would actually be quite straigtforward. And it does have structure: +: (%, %) -> % -- sum of two spaces *: (%, %) -> % -- intersection of two spaces 0: () -> % -- zero space Well, again, that's for later. Francois, could you please write a testcase for the functions, in the form testcase "columnSpace" assertEquals("columnSpace nicematrix", "result") assertEquals("columnSpace anothernicematrix", "anotheresult") Martin ------------------------------------------------------------------------- This SF.net email is sponsored by: Microsoft Defy all challenges. Microsoft(R) Visual Studio 2008. http://clk.atdmt.com/MRT/go/vse0120000070mrt/direct/01/ _______________________________________________ open-axiom-devel mailing list open-axiom-devel@lists.sourceforge.net https://lists.sourceforge.net/lists/listinfo/open-axiom-devel
