� ��g��<�ddlmZd d�Zdefd�Zd d�Zddd�d�ZdS) �)�_iszeroFc�T��|d��\}}�fd�|D��S)a�Returns 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 T��simplify�with_pivotsc�:��g|]}��|��S�)�col)�.0�i�Ms ��h/home/asafur/pinokio/api/open-webui.git/app/env/lib/python3.11/site-packages/sympy/matrices/subspaces.py� <listcomp>z _columnspace.<locals>.<listcomp>#s#��%�%�%��A�E�E�!�H�H�%�%�%�)�echelon_form)r r�reduced�pivotss` r�_columnspacers8��:�n�n�h�D�n�I�I�O�G�V�%�%�%�%�f�%�%�%�%rc�\�� ||��\}� � fd�t�j��D��}g}|D]^}�jg�jz}�j||<t� ��D]\}} || xx|||fzcc<�|�|��_�fd�|D��S)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 )� iszerofuncrc��g|]}|�v�|�� Sr r )rrrs �rrz_nullspace.<locals>.<listcomp>Bs��=�=�=�q�Q�f�_�_��_�_�_rc�H��g|]}��jd|��S)r)�_new�cols)r�br s �rrz_nullspace.<locals>.<listcomp>Ps+��0�0�0�Q�A�F�F�1�6�1�a� � �0�0�0r)�rref�ranger�zero�one� enumerate�append)r rrr� free_vars�basis�free_var�vec�piv_row�piv_colrs` @r� _nullspacer(&s��4�f�f� �X�f�F�F�O�G�V�=�=�=�=�E�!�&�M�M�=�=�=�I��E�� 1�6�)��H� � )�&� 1� 1� 7� 7��G�W��L�L�L�G�G�X�$5�6�6�L�L�L�L� ��S��0�0�0�0�%�0�0�0�0rc��|�|d��\�}�fd�tt|��D��S)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)rrrs �rrz_rowspace.<locals>.<listcomp>fs#��7�7�7�q�G�K�K��N�N�7�7�7r)rr�len)r rrrs @r� _rowspacer-SsF��"�n�n�h�D�n�I�I�O�G�V�7�7�7�7�E�#�f�+�+�$6�$6�7�7�7�7r)� normalize� rankcheckc��ddlm}|sgS|djdk}d�|D��}|j|�}|||��\}}|r'|jt|��krt d��g} t|j��D]I} |r||dd�| fj��}n||dd�| f��}| � |��J| S)a�Apply the Gram-Schmidt orthogonalization procedure to vectors supplied in ``vecs``. Parameters ========== vecs vectors to be made orthogonal normalize : bool If ``True``, return an orthonormal basis. rankcheck : bool If ``True``, the computation does not stop when encountering linearly dependent vectors. If ``False``, it will raise ``ValueError`` when any zero or linearly dependent vectors are found. Returns ======= list List of orthogonal (or orthonormal) basis vectors. Examples ======== >>> from sympy import I, Matrix >>> v = [Matrix([1, I]), Matrix([1, -I])] >>> Matrix.orthogonalize(*v) [Matrix([ [1], [I]]), Matrix([ [ 1], [-I]])] See Also ======== MatrixBase.QRdecomposition References ========== .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process r)�_QRdecomposition_optional�c�6�g|]}|��Sr )r%)r�xs rrz"_orthogonalize.<locals>.<listcomp>�s ��"�"�"��A�E�E�G�G�"�"�"r)r.z0GramSchmidt: vector set not linearly independentN) �decompositionsr1�rows�hstackrr,� ValueErrorr�Tr!)�clsr.r/�vecsr1�all_row_vecsr �Q�R�retrr s r�_orthogonalizer@is��`:�9�9�9�9�9�� G�L�A�%�L�"�"�T�"�"�"�D�� D��A�$�$�Q�)�<�<�<�D�A�q��M�Q�V�c�$�i�i�'�'��K�L�L�L� �C� �1�6�]�]�� #�a��1��g�i�.�.�C�C��#�a��1��g�,�,�C�� 3��JrN)F)� utilitiesrrr(r-r@r rr�<module>rBs��&�&�&�&�D!�W�*1�*1�*1�*1�Z8�8�8�8�,*/�%�E�E�E�E�E�E�Er