diff --git a/source/common/sagemath/library.sage b/source/common/sagemath/library.sage
index 2aa4e192a..4241ada9a 100644
--- a/source/common/sagemath/library.sage
+++ b/source/common/sagemath/library.sage
@@ -576,6 +576,18 @@ class TBIL:
string+=latex(self.vectors[-1])
return string
+ #Vector equation class
+ class LinearCombinationFromMatrix(LinearCombination):
+ def __init__(self,A,vars=None):
+ A=A.subdivision(0,0) # ignores augmented matrices
+ if vars is None:
+ self.coefficients=[var(f"x_{i}") for i in range(1,len(A.columns())+1)]
+ else:
+ self.coefficients=[vars[:len(A.columns())]]
+ self.vectors=[column_matrix(v) for v in A.columns()]
+ self.length=min(len(self.coefficients),len(self.vectors))
+ self.parentheses=False
+
#Generic equation, which could be used with polynomial or matrix equations. Often used with a LinearCombination passed as leftside
class Equation(SageObject):
def __init__(self,leftside,rightside):
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV1/generator.sage b/source/linear-algebra/exercises/outcomes/EV/EV1/generator.sage
index aee75e831..a498fd309 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV1/generator.sage
+++ b/source/linear-algebra/exercises/outcomes/EV/EV1/generator.sage
@@ -9,6 +9,10 @@ class Generator(BaseGenerator):
a,b,c,d = var("a b c d")
ls = [a,b,c,d][:rows]
+ # roll different statements
+ statements = [f"statement{l}" for l in "ABCDEF"]
+ shuffle(statements)
+
#start with nice RREF
number_of_pivots = 2
A = CheckIt.simple_random_matrix_of_rank(number_of_pivots,rows=rows,columns=columns)
@@ -32,14 +36,16 @@ class Generator(BaseGenerator):
for i in range(number_of_pivots)
],
)
- matrix = A.augment(column_matrix(lin_combo), subdivide=True)
+ A_aug = A.augment(column_matrix(lin_combo), subdivide=True)
vectors = [
{
"v": column_matrix(lin_combo),
"lin_combo": True,
"lin_combo_exp": lin_combo_exp,
- "A": matrix,
- "rref": matrix.rref(),
+ "A": A_aug,
+ "rref": A_aug.rref(),
+ "veceq": TBIL.VectorEquation(A_aug),
+ statements[0]: True,
}
]
@@ -53,13 +59,15 @@ class Generator(BaseGenerator):
choice([-1,1])
for _ in range(rows)
])
- matrix = A.augment(column_matrix(non_lin_combo), subdivide=True)
+ A_aug = A.augment(column_matrix(non_lin_combo), subdivide=True)
vectors += [
{
"v": column_matrix(non_lin_combo),
"lin_combo": False,
- "A": matrix,
- "rref": matrix.rref(),
+ "A": A_aug,
+ "rref": A_aug.rref(),
+ "veceq": TBIL.VectorEquation(A_aug),
+ statements[1]: True,
}
]
@@ -72,7 +80,6 @@ class Generator(BaseGenerator):
"vectors": vectors,
# "combovector": column_matrix(A.column(-1)),
# "statement": choice([True,False]),
- # "veceq": TBIL.VectorEquation(A),
# "matrix": A,
# "rref": A.rref(),
# "pivots": A.pivots(),
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV1/template.xml b/source/linear-algebra/exercises/outcomes/EV/EV1/template.xml
index b1404f907..22f08587a 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV1/template.xml
+++ b/source/linear-algebra/exercises/outcomes/EV/EV1/template.xml
@@ -16,7 +16,26 @@ that's equivalent to this claim.
- The vector equation (...) has at least one solution.
+ Here's an example of one such statement:
+
+ The vector equation {{veceq}} has at least one solution.
+
+
+ The following vector equation is consistent: {{veceq}}.
+
+
+ At least one solution exists for the equation {{veceq}}.
+
+
+ {{veceq}} has at least one solution vector.
+
+
+ A minimum of one solution for {{veceq}} can be found.
+
+
+ We can find at least one vector which solves the equation {{veceq}}.
+
+
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV2/generator.sage b/source/linear-algebra/exercises/outcomes/EV/EV2/generator.sage
index 82ee12f65..37a8e39e5 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV2/generator.sage
+++ b/source/linear-algebra/exercises/outcomes/EV/EV2/generator.sage
@@ -4,12 +4,18 @@ TBIL.config_matrix_typesetting()
class Generator(BaseGenerator):
def data(self):
A=CheckIt.simple_random_matrix_of_rank(choice([2,3]),rows=4,columns=3)
+
+ # roll different statements
+ statements = [f"statement{l}" for l in "ABCDEF"]
+ shuffle(statements)
tasks = [{
"spans": False,
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[0]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
spans = choice([True,False])
@@ -24,6 +30,8 @@ class Generator(BaseGenerator):
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[1]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
spans = not spans
@@ -38,6 +46,8 @@ class Generator(BaseGenerator):
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[2]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
shuffle(tasks)
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV2/template.xml b/source/linear-algebra/exercises/outcomes/EV/EV2/template.xml
index d33b3bef5..9623cae29 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV2/template.xml
+++ b/source/linear-algebra/exercises/outcomes/EV/EV2/template.xml
@@ -16,8 +16,25 @@
-
- The vector equation (...)=\vec w has at least one solution for every \vec w\in\mathbb R^4.
+
Here's an example of one such statement:
+
+ The vector equation {{veceqleft}}=\vec w has at least one solution for any \vec w\in\mathbb R^4.
+
+
+ For any \vec w\in\mathbb R^4, the following vector equation is consistent: {{veceqleft}}=\vec w.
+
+
+ At least one solution exists for the equation {{veceqleft}}=\vec w for every vector \vec w in \mathbb R^4.
+
+
+ {{veceqleft}}=\vec w has at least one solution vector regardless of how \vec w is chosen from \mathbb R^4.
+
+
+ Given any \vec w from \mathbb R^4, a minimum of one solution for {{veceqleft}}=\vec w can be found.
+
+
+ We can find at least one vector which solves the equation {{veceqleft}}=\vec w no matter what \vec w\in\mathbb R^4 is.
+
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV4/generator.sage b/source/linear-algebra/exercises/outcomes/EV/EV4/generator.sage
index 138e6631b..d86772742 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV4/generator.sage
+++ b/source/linear-algebra/exercises/outcomes/EV/EV4/generator.sage
@@ -4,12 +4,18 @@ TBIL.config_matrix_typesetting()
class Generator(BaseGenerator):
def data(self):
A=CheckIt.simple_random_matrix_of_rank(choice([3,4]),rows=4,columns=5)
+
+ # roll different statements
+ statements = [f"statement{l}" for l in "ABCDEF"]
+ shuffle(statements)
tasks = [{
"independent": False,
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[0]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
independent = choice([True,False])
@@ -24,6 +30,8 @@ class Generator(BaseGenerator):
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[1]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
independent = not independent
@@ -38,6 +46,8 @@ class Generator(BaseGenerator):
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[2]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
shuffle(tasks)
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV4/template.xml b/source/linear-algebra/exercises/outcomes/EV/EV4/template.xml
index f9c75ae35..eae27700e 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV4/template.xml
+++ b/source/linear-algebra/exercises/outcomes/EV/EV4/template.xml
@@ -16,8 +16,25 @@
-
- The vector equation (...)=\vec 0 has exactly one solution.
+
Here's an example of one such statement:
+
+ The vector equation {{veceqleft}}=\vec 0 has exactly one solution.
+
+
+ The following vector equation is consistent with a unique solution: {{veceqleft}}=\vec 0.
+
+
+ Exactly one solution exists for the equation {{veceqleft}}=\vec 0.
+
+
+ {{veceqleft}}=\vec 0 can only be solved by the solution vector \vec 0.
+
+
+ The unique solution for {{veceqleft}}=\vec 0 is the zero vector.
+
+
+ There is no vector besides \vec 0 which solves the equation {{veceqleft}}=\vec 0.
+
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV5/generator.sage b/source/linear-algebra/exercises/outcomes/EV/EV5/generator.sage
index f9302b8bc..5d2c6b210 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV5/generator.sage
+++ b/source/linear-algebra/exercises/outcomes/EV/EV5/generator.sage
@@ -4,12 +4,18 @@ TBIL.config_matrix_typesetting()
class Generator(BaseGenerator):
def data(self):
A=CheckIt.simple_random_matrix_of_rank(choice([2,3]),rows=4,columns=choice([3,5]))
+
+ # roll different statements
+ statements = [f"statement{l}" for l in "ABCDEF"]
+ shuffle(statements)
tasks = [{
"basis": False,
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[0]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
basis = choice([True,False])
@@ -24,6 +30,8 @@ class Generator(BaseGenerator):
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[1]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
basis = not basis
@@ -38,6 +46,8 @@ class Generator(BaseGenerator):
"vecset": TBIL.VectorSet(A.columns()),
"matrix": A,
"rref": A.rref(),
+ statements[2]: True,
+ "veceqleft": TBIL.LinearCombinationFromMatrix(A),
}]
shuffle(tasks)
diff --git a/source/linear-algebra/exercises/outcomes/EV/EV5/template.xml b/source/linear-algebra/exercises/outcomes/EV/EV5/template.xml
index 151335ce0..ee38aa9d3 100644
--- a/source/linear-algebra/exercises/outcomes/EV/EV5/template.xml
+++ b/source/linear-algebra/exercises/outcomes/EV/EV5/template.xml
@@ -16,8 +16,25 @@
-
- The vector equation (...)=\vec w has exactly one solution for every \vec w\in\mathbb R^4.
+
Here's an example of one such statement:
+
+ The vector equation {{veceqleft}}=\vec w has a unique solution for any \vec w\in\mathbb R^4.
+
+
+ For any \vec w\in\mathbb R^4, the following vector equation has exactly one solution: {{veceqleft}}=\vec w.
+
+
+ One and only one solution exists for the equation {{veceqleft}}=\vec w for every vector \vec w in \mathbb R^4.
+
+
+ {{veceqleft}}=\vec w has a unique solution regardless of how \vec w is chosen from \mathbb R^4.
+
+
+ Given any \vec w from \mathbb R^4, exactly one solution for {{veceqleft}}=\vec w can be found.
+
+
+ We can find a unique vector which solves the equation {{veceqleft}}=\vec w no matter what \vec w\in\mathbb R^4 is.
+