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jkostiuk merged 10 commits into from
Feb 25, 2026
Merged

add more statements to EV1, EV2, EV4, EV5 #940

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0ab680b
add more statements to EV1
StevenClontz Feb 25, 2026
ceed8e9
add equations for EV2
StevenClontz Feb 25, 2026
2bd5b90
comment in LinearCombinationFromMatrix
StevenClontz Feb 25, 2026
f1fd4b7
add statements for EV4, EV5
StevenClontz Feb 25, 2026
c3d2322
Update source/linear-algebra/exercises/outcomes/EV/EV2/template.xml
StevenClontz Feb 25, 2026
b6a70da
Update source/linear-algebra/exercises/outcomes/EV/EV2/template.xml
StevenClontz Feb 25, 2026
0ba24b0
Update source/linear-algebra/exercises/outcomes/EV/EV4/template.xml
StevenClontz Feb 25, 2026
c8e24b7
Update source/linear-algebra/exercises/outcomes/EV/EV5/template.xml
StevenClontz Feb 25, 2026
5fb2b88
Update source/linear-algebra/exercises/outcomes/EV/EV5/template.xml
StevenClontz Feb 25, 2026
88903fc
elaborate that we just give one such example
StevenClontz Feb 25, 2026
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12 changes: 12 additions & 0 deletions source/common/sagemath/library.sage
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Expand Up @@ -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):
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21 changes: 14 additions & 7 deletions source/linear-algebra/exercises/outcomes/EV/EV1/generator.sage
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Original file line number Diff line number Diff line change
Expand Up @@ -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)
Expand All @@ -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,
}
]

Expand All @@ -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,
}
]

Expand All @@ -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(),
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21 changes: 20 additions & 1 deletion source/linear-algebra/exercises/outcomes/EV/EV1/template.xml
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Expand Up @@ -16,7 +16,26 @@ that's equivalent to this claim.
</p>
</content>
<outtro>
<p><q>The vector equation (...) has at least one solution.</q></p>
<p>Here's an example of one such statement:
<!-- {{#statementA}} -->
<q>The vector equation <m>{{veceq}}</m> has at least one solution.</q>
<!-- {{/statementA}} -->
<!-- {{#statementB}} -->
<q>The following vector equation is consistent: <m>{{veceq}}</m>.</q>
<!-- {{/statementB}} -->
<!-- {{#statementC}} -->
<q>At least one solution exists for the equation <m>{{veceq}}</m>.</q>
<!-- {{/statementC}} -->
<!-- {{#statementD}} -->
<q><m>{{veceq}}</m> has at least one solution vector.</q>
<!-- {{/statementD}} -->
<!-- {{#statementE}} -->
<q>A minimum of one solution for <m>{{veceq}}</m> can be found.</q>
<!-- {{/statementE}} -->
<!-- {{#statementF}} -->
<q>We can find at least one vector which solves the equation <m>{{veceq}}</m>.</q>
<!-- {{/statementF}} -->
</p>
</outtro>
</knowl>
<knowl>
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10 changes: 10 additions & 0 deletions source/linear-algebra/exercises/outcomes/EV/EV2/generator.sage
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Original file line number Diff line number Diff line change
Expand Up @@ -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])
Expand All @@ -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
Expand All @@ -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)
Expand Down
21 changes: 19 additions & 2 deletions source/linear-algebra/exercises/outcomes/EV/EV2/template.xml
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Expand Up @@ -16,8 +16,25 @@
</p>
</content>
<outtro>
<p>
<q>The vector equation <m>(...)=\vec w</m> has at least one solution for every <m>\vec w\in\mathbb R^4</m>.</q>
<p>Here's an example of one such statement:
<!-- {{#statementA}} -->
<q>The vector equation <m>{{veceqleft}}=\vec w</m> has at least one solution for any <m>\vec w\in\mathbb R^4</m>.</q>
<!-- {{/statementA}} -->
<!-- {{#statementB}} -->
<q>For any <m>\vec w\in\mathbb R^4</m>, the following vector equation is consistent: <m>{{veceqleft}}=\vec w</m>.</q>
<!-- {{/statementB}} -->
<!-- {{#statementC}} -->
<q>At least one solution exists for the equation <m>{{veceqleft}}=\vec w</m> for every vector <m>\vec w</m> in <m>\mathbb R^4</m>.</q>
<!-- {{/statementC}} -->
<!-- {{#statementD}} -->
<q><m>{{veceqleft}}=\vec w</m> has at least one solution vector regardless of how <m>\vec w</m> is chosen from <m>\mathbb R^4</m>.</q>
<!-- {{/statementD}} -->
<!-- {{#statementE}} -->
<q>Given any <m>\vec w</m> from <m>\mathbb R^4</m>, a minimum of one solution for <m>{{veceqleft}}=\vec w</m> can be found.</q>
<!-- {{/statementE}} -->
<!-- {{#statementF}} -->
<q>We can find at least one vector which solves the equation <m>{{veceqleft}}=\vec w</m> no matter what <m>\vec w\in\mathbb R^4</m> is.</q>
<!-- {{/statementF}} -->
</p>
</outtro>
</knowl>
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10 changes: 10 additions & 0 deletions source/linear-algebra/exercises/outcomes/EV/EV4/generator.sage
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Expand Up @@ -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])
Expand All @@ -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
Expand All @@ -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)
Expand Down
21 changes: 19 additions & 2 deletions source/linear-algebra/exercises/outcomes/EV/EV4/template.xml
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Expand Up @@ -16,8 +16,25 @@
</p>
</content>
<outtro>
<p>
<q>The vector equation <m>(...)=\vec 0</m> has exactly one solution.</q>
<p>Here's an example of one such statement:
<!-- {{#statementA}} -->
<q>The vector equation <m>{{veceqleft}}=\vec 0</m> has exactly one solution.</q>
<!-- {{/statementA}} -->
<!-- {{#statementB}} -->
<q>The following vector equation is consistent with a unique solution: <m>{{veceqleft}}=\vec 0</m>.</q>
<!-- {{/statementB}} -->
<!-- {{#statementC}} -->
<q>Exactly one solution exists for the equation <m>{{veceqleft}}=\vec 0</m>.</q>
<!-- {{/statementC}} -->
<!-- {{#statementD}} -->
<q><m>{{veceqleft}}=\vec 0</m> can only be solved by the solution vector <m>\vec 0</m>.</q>
<!-- {{/statementD}} -->
<!-- {{#statementE}} -->
<q>The unique solution for <m>{{veceqleft}}=\vec 0</m> is the zero vector.</q>
<!-- {{/statementE}} -->
<!-- {{#statementF}} -->
<q>There is no vector besides <m>\vec 0</m> which solves the equation <m>{{veceqleft}}=\vec 0</m>.</q>
<!-- {{/statementF}} -->
</p>
</outtro>
</knowl>
Expand Down
10 changes: 10 additions & 0 deletions source/linear-algebra/exercises/outcomes/EV/EV5/generator.sage
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Original file line number Diff line number Diff line change
Expand Up @@ -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])
Expand All @@ -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
Expand All @@ -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)
Expand Down
21 changes: 19 additions & 2 deletions source/linear-algebra/exercises/outcomes/EV/EV5/template.xml
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Expand Up @@ -16,8 +16,25 @@
</p>
</content>
<outtro>
<p>
<q>The vector equation <m>(...)=\vec w</m> has exactly one solution for every <m>\vec w\in\mathbb R^4</m>.</q>
<p>Here's an example of one such statement:
<!-- {{#statementA}} -->
<q>The vector equation <m>{{veceqleft}}=\vec w</m> has a unique solution for any <m>\vec w\in\mathbb R^4</m>.</q>
<!-- {{/statementA}} -->
<!-- {{#statementB}} -->
<q>For any <m>\vec w\in\mathbb R^4</m>, the following vector equation has exactly one solution: <m>{{veceqleft}}=\vec w</m>.</q>
<!-- {{/statementB}} -->
<!-- {{#statementC}} -->
<q>One and only one solution exists for the equation <m>{{veceqleft}}=\vec w</m> for every vector <m>\vec w</m> in <m>\mathbb R^4</m>.</q>
<!-- {{/statementC}} -->
<!-- {{#statementD}} -->
<q><m>{{veceqleft}}=\vec w</m> has a unique solution regardless of how <m>\vec w</m> is chosen from <m>\mathbb R^4</m>.</q>
<!-- {{/statementD}} -->
<!-- {{#statementE}} -->
<q>Given any <m>\vec w</m> from <m>\mathbb R^4</m>, exactly one solution for <m>{{veceqleft}}=\vec w</m> can be found.</q>
<!-- {{/statementE}} -->
<!-- {{#statementF}} -->
<q>We can find a unique vector which solves the equation <m>{{veceqleft}}=\vec w</m> no matter what <m>\vec w\in\mathbb R^4</m> is.</q>
<!-- {{/statementF}} -->
</p>
</outtro>
</knowl>
Expand Down