*hBSSKJr S SjrS\4SjrS SjrSSS.Sjrg) )_iszeroFcrURUSS9up#UVs/sHo@RU5PM sn$s snf)aYReturns a list of vectors (Matrix objects) that span columnspace of ``M`` Examples ======== >>> from sympy import Matrix >>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) >>> M Matrix([ [ 1, 3, 0], [-2, -6, 0], [ 3, 9, 6]]) >>> M.columnspace() [Matrix([ [ 1], [-2], [ 3]]), Matrix([ [0], [0], [6]])] See Also ======== nullspace rowspace Tsimplify with_pivots) echelon_formcolMrreducedpivotsis U/home/mande/repo/quber/.venv/lib/python3.13/site-packages/sympy/matrices/subspaces.py _columnspacers6:nnhDnIOG$ %fEE!Hf %% %s4cURX!S9up4[UR5Vs/sH oUU;dM UPM nn/nUH`nUR/UR-n URX'[ U5HupX==X:U4-ss'M UR U 5 Mb UV s/sHoRURSU 5PM! sn $s snfs sn f)a;Returns list of vectors (Matrix objects) that span nullspace of ``M`` Examples ======== >>> from sympy import Matrix >>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) >>> M Matrix([ [ 1, 3, 0], [-2, -6, 0], [ 3, 9, 6]]) >>> M.nullspace() [Matrix([ [-3], [ 1], [ 0]])] See Also ======== columnspace rowspace ) iszerofuncrr)rrefrangecolszeroone enumerateappend_new) r rrr r r free_varsbasisfree_varvecpiv_rowpiv_colbs r _nullspacer"&s4ff fFOG!!&&M=Mqf_MI=E166) )& 1 G LGX$56 6L!2  S+0 0%QFF1661a % 00> 1s CC*&CcURUSS9up#[[U55Vs/sHoBRU5PM sn$s snf)aReturns a list of vectors that span the row space of ``M``. Examples ======== >>> from sympy import Matrix >>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) >>> M Matrix([ [ 1, 3, 0], [-2, -6, 0], [ 3, 9, 6]]) >>> M.rowspace() [Matrix([[1, 3, 0]]), Matrix([[0, 0, 6]])] Tr)rrlenrowr s r _rowspacer&SsA"nnhDnIOG$)#f+$6 7$6qKKN$6 77 7sA) normalize rankcheckcSSKJn U(d/$USRS:HnUVs/sHofR5PM nnUR"U6nU"XqS9upU(a$UR [ U5:a [S5e/n [UR 5HDn U(aU"USS2U 4R5n OU"USS2U 45n U RU 5 MF U $s snf)a0Apply the Gram-Schmidt orthogonalization procedure to vectors supplied in ``vecs``. Parameters ========== vecs vectors to be made orthogonal normalize : bool If ``True``, return an orthonormal basis. rankcheck : bool If ``True``, the computation does not stop when encountering linearly dependent vectors. If ``False``, it will raise ``ValueError`` when any zero or linearly dependent vectors are found. Returns ======= list List of orthogonal (or orthonormal) basis vectors. Examples ======== >>> from sympy import I, Matrix >>> v = [Matrix([1, I]), Matrix([1, -I])] >>> Matrix.orthogonalize(*v) [Matrix([ [1], [I]]), Matrix([ [ 1], [-I]])] See Also ======== MatrixBase.QRdecomposition References ========== .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process r)_QRdecomposition_optional)r'z0GramSchmidt: vector set not linearly independentN) decompositionsr*rowsrhstackrr$ ValueErrorrTr) clsr'r(vecsr* all_row_vecsxr QRretrr s r_orthogonalizer8is`:  GLLA%L! "TEEGTD " DA $Q r<s-&D!W*1Z8,*/%Er;