Ed<ddlmZd dZdefdZd dZddddZdS) )_iszeroFcT|d\}}fd|DS)aReturns 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_pivotsc:g|]}|S)col).0iMs 8lib/python3.11/site-packages/sympy/matrices/subspaces.py z _columnspace..#s# % % %AEE!HH % % %) echelon_form)r rreducedpivotss` r _columnspacers8:nnhDnIIOGV % % % %f % % %%rc\ ||\} fdtjD}g}|D]^}jgjz}j||<t D]\}} || xx|||fzcc<||_fd|DS)aReturns 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 ) iszerofuncrcg|]}|v| Sr r )r r rs rrz_nullspace..Bs"===qQf_====rcHg|]}jd|S)r)_newcols)r br s rrz_nullspace..Ps+ 0 0 0QAFF161a 0 0 0r)rrefrangerzeroone enumerateappend) r rrr free_varsbasisfree_varvecpiv_rowpiv_colrs ` @r _nullspacer(&s4ff XfFFOGV====E!&MM===IE  16)H )& 1 1 7 7 GW LLLGGX$56 6LLLL S 0 0 0 0% 0 0 00rc||d\}fdtt|DS)aDReturns 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]])] Trc:g|]}|Sr )row)r r rs rrz_rowspace..fs# 7 7 7qGKKNN 7 7 7r)rrlen)r rrrs @r _rowspacer-SsF"nnhDnIIOGV 7 7 7 7E#f++$6$6 7 7 77r) normalize rankcheckcddlm}|sgS|djdk}d|D}|j|}|||\}}|r'|jt |krt dg} t|jD]I} |r||dd| fj} n||dd| f} | | J| S)aApply 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_optionalc6g|]}|Sr )r%)r xs rrz"_orthogonalize..s " " "AEEGG " " "r)r.z0GramSchmidt: vector set not linearly independentN) decompositionsr1rowshstackrr, ValueErrorrTr!) clsr.r/vecsr1 all_row_vecsr QRretr r s r_orthogonalizer@is `:99999  GLA%L " "T " " "D DA $ $Q) < < rBs&&&&D!W*1*1*1*1Z8888,*/%EEEEEEEr