o o h@sBddlmZd ddZdefddZd ddZddd d d Zd S))_iszeroFcs$j|dd\}}fdd|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_pivotscg|]}|qS)col.0iMrl/var/www/html/construction_image-detection-poc/venv/lib/python3.10/site-packages/sympy/matrices/subspaces.py #z _columnspace..) echelon_form)r rreducedpivotsrr r _columnspacesrc sj||d\}fddtjD}g}|D](}jgj}j||<tD]\}} || |||f8<q+||qfdd|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 ) iszerofuncrcsg|]}|vr|qSrrr )rrrrBsz_nullspace..csg|] }jd|qS)r)_newcols)r br rrrPs)rrefrangerzeroone enumerateappend) r rrr free_varsbasisfree_varvecpiv_rowpiv_colr)r rr _nullspace&s  r%cs,|j|dd\}fddtt|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]])] Trcrr)rowr rrrrfrz_rowspace..)rrlen)r rrrr'r _rowspaceSsr)) normalize rankcheckc Gsddlm}|s gS|djdk}dd|D}|j|}|||d\}}|r2|jt|kr2tdg} t|jD]} |rI||dd| fj} n ||dd| f} | | q9| 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_optionalcSsg|]}|qSr)r")r xrrrrsz"_orthogonalize..)r*z0GramSchmidt: vector set not linearly independentN) decompositionsr,rowshstackrr( ValueErrorrTr) clsr*r+vecsr, all_row_vecsr QRretr rrrr_orthogonalizeis 0  r:N)F) utilitiesrrr%r)r:rrrrs " -