From e270655ec552abb098a27cb61f3d78ddd2a1886a Mon Sep 17 00:00:00 2001
From: Travis Scrimshaw <tcscrims@gmail.com>
Date: Mon, 21 Jun 2021 09:21:14 +1000
Subject: [PATCH] Fixing issue when the support is smaller than the dimension.

---
 .../finite_dimensional_modules_with_basis.py   | 17 ++++++++++++++++-
 src/sage/matrix/matrix_space.py                | 18 ++++++++++++++++--
 2 files changed, 32 insertions(+), 3 deletions(-)

diff --git a/src/sage/categories/finite_dimensional_modules_with_basis.py b/src/sage/categories/finite_dimensional_modules_with_basis.py
index c4642d2fbb3..9be638117b7 100644
--- a/src/sage/categories/finite_dimensional_modules_with_basis.py
+++ b/src/sage/categories/finite_dimensional_modules_with_basis.py
@@ -336,6 +336,17 @@ def echelon_form(self, elements, row_reduced=False, order=None):
                 sage: x = C.basis()
                 sage: C.echelon_form([x[0] - x[1], 2*x[1] - 2*x[2], x[0] - x[2]])
                 [x[0] - x[2], x[1] - x[2]]
+
+            ::
+
+                sage: M = MatrixSpace(QQ, 3, 3)                                                                                     
+                sage: A = M([[0, 0, 2], [0, 0, 0], [0, 0, 0]])                                                                      
+                sage: M.echelon_form([A, A])                                                                                         
+                [
+                [0 0 1]
+                [0 0 0]
+                [0 0 0]
+                ]
             """
             if order is not None:
                 order = self._compute_support_order(elements, order)
@@ -394,7 +405,11 @@ def _vector_(self, order=None):
                 sage: C.an_element()._vector_()
                 (2, 2, 3)
             """
-            dense_free_module = self.parent()._dense_free_module()
+            if order is None:
+                dense_free_module = self.parent()._dense_free_module()
+            else:
+                from sage.modules.free_module import FreeModule
+                dense_free_module = FreeModule(self.parent().base_ring(), len(order))
             # We slightly break encapsulation for speed reasons
             return dense_free_module.element_class(dense_free_module,
                                                    self.dense_coefficient_list(order),
diff --git a/src/sage/matrix/matrix_space.py b/src/sage/matrix/matrix_space.py
index 3d9cf880460..8029c6a08b1 100644
--- a/src/sage/matrix/matrix_space.py
+++ b/src/sage/matrix/matrix_space.py
@@ -1586,7 +1586,7 @@ def submodule(self, gens, check=True, already_echelonized=False,
 
         If ``already_echelonized`` is ``False``, then the
         generators are put in reduced echelon form using
-        :meth:`echelonize`, and reindexed by `0,1,...`.
+        :meth:`echelonize`, and reindexed by `0, 1, \ldots`.
 
         .. WARNING::
 
@@ -1601,7 +1601,6 @@ def submodule(self, gens, check=True, already_echelonized=False,
         The basis of the submodule uses the same index set as the
         generators, and the lifting map sends `y_i` to `gens[i]`.
 
-
         .. SEEALSO::
 
              :meth:`ModulesWithBasis.ParentMethods.submodule`
@@ -1620,6 +1619,21 @@ def submodule(self, gens, check=True, already_echelonized=False,
             [ 1  0]  [0 1]
             [-3  2], [3 1]
             ]
+
+            sage: A = matrix([[1, 1], [0, -1]])
+            sage: B = matrix([[0, 1], [0, 2]])
+            sage: X = M.submodule([A, B])
+            sage: Xp = M.submodule([A, B], order=[(0,1), (1,1), (0,0)])
+            sage: [X.lift(b) for b in Xp.basis()]
+            [
+            [ 1  0]  [0 1]
+            [ 0 -3], [0 2]
+            ]
+            sage: [Xp.lift(b) for b in X.basis()]
+            [
+            [2/3   1]  [-1/3    0]
+            [  0   0], [   0    1]
+            ]
         """
         support_order = self._compute_support_order(gens, support_order)
         if not already_echelonized: