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Sage B Bases

Every vector space in Sage has a basis — you can obtain this with the vector space method .basis(), and the result is a list of vectors. Another method for a vector space is .basis_matrix() which outputs a matrix whose rows are the vectors of a basis. Sometimes one form is more convenient that the other, but notice that the description of a vector space chooses to print the basis matrix (since its display is just a bit easier to read). A vector space typically has many bases (infinitely many), so which one does Sage use? You will notice that the basis matrices displayed are in reduced row-echelon form — this is the defining property of the basis chosen by Sage.

Here is Example RSB again as an example of how bases are provided in Sage.