+ iF^RIHtR RltR]3RltR RltRRRR/RltR #) )_iszeroFcvVPVRR7wr#VUu.uFq@PV4NK up#uupi)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&& x/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/matrices/subspaces.py _columnspacers6:nnhDnIOG$ %fEE!Hf %% %s6cVPW!R7wr4\VP4Uu.uF qUV9gK VNK pp.pVFppVP.VP,p VPW&\ V4F!wrW;;,W:V3,,uu&K# VP V 4Kr VU u.uFqPVP^V 4NK! up #uupiuup i)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 ) iszerofuncr)rrefrangecolszeroone enumerateappend_new) r rrr r r free_varsbasisfree_varvecpiv_rowpiv_colbs &&& r _nullspacer"&s4ff fFOG!!&&M=Mqf_MI=E166) )& 1 G LGX$56 6L!2  S+0 0%QFF1661a % 00> 1sC%C%=%C*cVPVRR7wr#\\V44Uu.uFqBPV4NK up#uupi)aReturns a list of vectors that span the row space of ``M``. Examples ======== >>> from sympy import Matrix >>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) >>> M Matrix([ [ 1, 3, 0], [-2, -6, 0], [ 3, 9, 6]]) >>> M.rowspace() [Matrix([[1, 3, 0]]), Matrix([[0, 0, 6]])] Tr)rrlenrowr s&& r _rowspacer&SsA"nnhDnIOG$)#f+$6 7$6qKKN$6 77 7sA normalize rankcheckc^RIHpV'g.#V^,P^8HpVUu.uFqfP4NK ppVP!V!pV!WqR7wrV'd&VP \ V48d \R4h.p \VP 4FIp V'dV!VRV 3,P4p MV!VRV 3,4p V PV 4KK V #uupi)a0Apply the Gram-Schmidt orthogonalization procedure to vectors supplied in ``vecs``. Parameters ========== vecs vectors to be made orthogonal normalize : bool If ``True``, return an orthonormal basis. rankcheck : bool If ``True``, the computation does not stop when encountering linearly dependent vectors. If ``False``, it will raise ``ValueError`` when any zero or linearly dependent vectors are found. Returns ======= list List of orthogonal (or orthonormal) basis vectors. Examples ======== >>> from sympy import I, Matrix >>> v = [Matrix([1, I]), Matrix([1, -I])] >>> Matrix.orthogonalize(*v) [Matrix([ [1], [I]]), Matrix([ [ 1], [-I]])] See Also ======== MatrixBase.QRdecomposition References ========== .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process )_QRdecomposition_optional)r'z0GramSchmidt: vector set not linearly independent:NNN) decompositionsr*rowsrhstackrr$ ValueErrorrTr) clsr'r(vecsr* all_row_vecsxr QRretrr s &$$* r_orthogonalizer7is`:  GLLA%L! "TEEGTD " DA $Q r;s6&D!W*1Z8,EE%Er: