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StevenClontzsiwelwerd
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add more statements to EV1, EV2, EV4, EV5 (#940)
* add more statements to EV1 * add equations for EV2 * comment in LinearCombinationFromMatrix * add statements for EV4, EV5 * Update source/linear-algebra/exercises/outcomes/EV/EV2/template.xml Co-authored-by: Drew Lewis <30658947+siwelwerd@users.noreply.github.com> * Update source/linear-algebra/exercises/outcomes/EV/EV2/template.xml Co-authored-by: Drew Lewis <30658947+siwelwerd@users.noreply.github.com> * Update source/linear-algebra/exercises/outcomes/EV/EV4/template.xml Co-authored-by: Drew Lewis <30658947+siwelwerd@users.noreply.github.com> * Update source/linear-algebra/exercises/outcomes/EV/EV5/template.xml Co-authored-by: Drew Lewis <30658947+siwelwerd@users.noreply.github.com> * Update source/linear-algebra/exercises/outcomes/EV/EV5/template.xml Co-authored-by: Drew Lewis <30658947+siwelwerd@users.noreply.github.com> * elaborate that we just give one such example --------- Co-authored-by: Drew Lewis <30658947+siwelwerd@users.noreply.github.com>
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9 files changed

Lines changed: 133 additions & 14 deletions

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‎source/common/sagemath/library.sage‎

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@@ -576,6 +576,18 @@ class TBIL:
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string+=latex(self.vectors[-1])
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return string
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#Vector equation class
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class LinearCombinationFromMatrix(LinearCombination):
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def __init__(self,A,vars=None):
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A=A.subdivision(0,0) # ignores augmented matrices
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if vars is None:
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self.coefficients=[var(f"x_{i}") for i in range(1,len(A.columns())+1)]
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else:
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self.coefficients=[vars[:len(A.columns())]]
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self.vectors=[column_matrix(v) for v in A.columns()]
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self.length=min(len(self.coefficients),len(self.vectors))
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self.parentheses=False
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#Generic equation, which could be used with polynomial or matrix equations. Often used with a LinearCombination passed as leftside
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class Equation(SageObject):
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def __init__(self,leftside,rightside):

‎source/linear-algebra/exercises/outcomes/EV/EV1/generator.sage‎

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@@ -9,6 +9,10 @@ class Generator(BaseGenerator):
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a,b,c,d = var("a b c d")
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ls = [a,b,c,d][:rows]
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# roll different statements
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statements = [f"statement{l}" for l in "ABCDEF"]
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shuffle(statements)
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#start with nice RREF
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number_of_pivots = 2
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A = CheckIt.simple_random_matrix_of_rank(number_of_pivots,rows=rows,columns=columns)
@@ -32,14 +36,16 @@ class Generator(BaseGenerator):
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for i in range(number_of_pivots)
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],
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)
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matrix = A.augment(column_matrix(lin_combo), subdivide=True)
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A_aug = A.augment(column_matrix(lin_combo), subdivide=True)
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vectors = [
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{
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"v": column_matrix(lin_combo),
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"lin_combo": True,
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"lin_combo_exp": lin_combo_exp,
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"A": matrix,
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"rref": matrix.rref(),
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"A": A_aug,
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"rref": A_aug.rref(),
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"veceq": TBIL.VectorEquation(A_aug),
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statements[0]: True,
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}
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]
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@@ -53,13 +59,15 @@ class Generator(BaseGenerator):
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choice([-1,1])
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for _ in range(rows)
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])
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matrix = A.augment(column_matrix(non_lin_combo), subdivide=True)
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A_aug = A.augment(column_matrix(non_lin_combo), subdivide=True)
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vectors += [
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{
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"v": column_matrix(non_lin_combo),
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"lin_combo": False,
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"A": matrix,
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"rref": matrix.rref(),
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"A": A_aug,
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"rref": A_aug.rref(),
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"veceq": TBIL.VectorEquation(A_aug),
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statements[1]: True,
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}
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]
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@@ -72,7 +80,6 @@ class Generator(BaseGenerator):
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"vectors": vectors,
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# "combovector": column_matrix(A.column(-1)),
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# "statement": choice([True,False]),
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# "veceq": TBIL.VectorEquation(A),
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# "matrix": A,
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# "rref": A.rref(),
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# "pivots": A.pivots(),

‎source/linear-algebra/exercises/outcomes/EV/EV1/template.xml‎

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@@ -16,7 +16,26 @@ that's equivalent to this claim.
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</p>
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</content>
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<outtro>
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<p><q>The vector equation (...) has at least one solution.</q></p>
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<p>Here's an example of one such statement:
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<!-- {{#statementA}} -->
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<q>The vector equation <m>{{veceq}}</m> has at least one solution.</q>
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<!-- {{/statementA}} -->
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<!-- {{#statementB}} -->
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<q>The following vector equation is consistent: <m>{{veceq}}</m>.</q>
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<!-- {{/statementB}} -->
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<!-- {{#statementC}} -->
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<q>At least one solution exists for the equation <m>{{veceq}}</m>.</q>
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<!-- {{/statementC}} -->
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<!-- {{#statementD}} -->
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<q><m>{{veceq}}</m> has at least one solution vector.</q>
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<!-- {{/statementD}} -->
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<!-- {{#statementE}} -->
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<q>A minimum of one solution for <m>{{veceq}}</m> can be found.</q>
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<!-- {{/statementE}} -->
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<!-- {{#statementF}} -->
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<q>We can find at least one vector which solves the equation <m>{{veceq}}</m>.</q>
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<!-- {{/statementF}} -->
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</p>
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</outtro>
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</knowl>
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<knowl>

‎source/linear-algebra/exercises/outcomes/EV/EV2/generator.sage‎

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@@ -4,12 +4,18 @@ TBIL.config_matrix_typesetting()
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class Generator(BaseGenerator):
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def data(self):
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A=CheckIt.simple_random_matrix_of_rank(choice([2,3]),rows=4,columns=3)
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# roll different statements
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statements = [f"statement{l}" for l in "ABCDEF"]
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shuffle(statements)
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tasks = [{
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"spans": False,
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[0]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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spans = choice([True,False])
@@ -24,6 +30,8 @@ class Generator(BaseGenerator):
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[1]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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spans = not spans
@@ -38,6 +46,8 @@ class Generator(BaseGenerator):
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[2]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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shuffle(tasks)

‎source/linear-algebra/exercises/outcomes/EV/EV2/template.xml‎

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Original file line numberDiff line numberDiff line change
@@ -16,8 +16,25 @@
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</p>
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</content>
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<outtro>
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<p>
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<q>The vector equation <m>(...)=\vec w</m> has at least one solution for every <m>\vec w\in\mathbb R^4</m>.</q>
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<p>Here's an example of one such statement:
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<!-- {{#statementA}} -->
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<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>
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<!-- {{/statementA}} -->
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<!-- {{#statementB}} -->
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<q>For any <m>\vec w\in\mathbb R^4</m>, the following vector equation is consistent: <m>{{veceqleft}}=\vec w</m>.</q>
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<!-- {{/statementB}} -->
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<!-- {{#statementC}} -->
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<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>
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<!-- {{/statementC}} -->
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<!-- {{#statementD}} -->
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<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>
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<!-- {{/statementD}} -->
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<!-- {{#statementE}} -->
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<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>
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<!-- {{/statementE}} -->
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<!-- {{#statementF}} -->
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<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>
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<!-- {{/statementF}} -->
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</p>
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</outtro>
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</knowl>

‎source/linear-algebra/exercises/outcomes/EV/EV4/generator.sage‎

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@@ -4,12 +4,18 @@ TBIL.config_matrix_typesetting()
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class Generator(BaseGenerator):
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def data(self):
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A=CheckIt.simple_random_matrix_of_rank(choice([3,4]),rows=4,columns=5)
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# roll different statements
9+
statements = [f"statement{l}" for l in "ABCDEF"]
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shuffle(statements)
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tasks = [{
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"independent": False,
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[0]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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independent = choice([True,False])
@@ -24,6 +30,8 @@ class Generator(BaseGenerator):
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[1]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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independent = not independent
@@ -38,6 +46,8 @@ class Generator(BaseGenerator):
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[2]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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shuffle(tasks)

‎source/linear-algebra/exercises/outcomes/EV/EV4/template.xml‎

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@@ -16,8 +16,25 @@
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</p>
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</content>
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<outtro>
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<p>
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<q>The vector equation <m>(...)=\vec 0</m> has exactly one solution.</q>
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<p>Here's an example of one such statement:
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<!-- {{#statementA}} -->
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<q>The vector equation <m>{{veceqleft}}=\vec 0</m> has exactly one solution.</q>
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<!-- {{/statementA}} -->
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<!-- {{#statementB}} -->
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<q>The following vector equation is consistent with a unique solution: <m>{{veceqleft}}=\vec 0</m>.</q>
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<!-- {{/statementB}} -->
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<!-- {{#statementC}} -->
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<q>Exactly one solution exists for the equation <m>{{veceqleft}}=\vec 0</m>.</q>
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<!-- {{/statementC}} -->
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<!-- {{#statementD}} -->
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<q><m>{{veceqleft}}=\vec 0</m> can only be solved by the solution vector <m>\vec 0</m>.</q>
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<!-- {{/statementD}} -->
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<!-- {{#statementE}} -->
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<q>The unique solution for <m>{{veceqleft}}=\vec 0</m> is the zero vector.</q>
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<!-- {{/statementE}} -->
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<!-- {{#statementF}} -->
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<q>There is no vector besides <m>\vec 0</m> which solves the equation <m>{{veceqleft}}=\vec 0</m>.</q>
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<!-- {{/statementF}} -->
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</p>
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</outtro>
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</knowl>

‎source/linear-algebra/exercises/outcomes/EV/EV5/generator.sage‎

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@@ -4,12 +4,18 @@ TBIL.config_matrix_typesetting()
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class Generator(BaseGenerator):
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def data(self):
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A=CheckIt.simple_random_matrix_of_rank(choice([2,3]),rows=4,columns=choice([3,5]))
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# roll different statements
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statements = [f"statement{l}" for l in "ABCDEF"]
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shuffle(statements)
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812
tasks = [{
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"basis": False,
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[0]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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1521
basis = choice([True,False])
@@ -24,6 +30,8 @@ class Generator(BaseGenerator):
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"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[1]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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2937
basis = not basis
@@ -38,6 +46,8 @@ class Generator(BaseGenerator):
3846
"vecset": TBIL.VectorSet(A.columns()),
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"matrix": A,
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"rref": A.rref(),
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statements[2]: True,
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"veceqleft": TBIL.LinearCombinationFromMatrix(A),
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}]
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shuffle(tasks)

‎source/linear-algebra/exercises/outcomes/EV/EV5/template.xml‎

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@@ -16,8 +16,25 @@
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</p>
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</content>
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<outtro>
19-
<p>
20-
<q>The vector equation <m>(...)=\vec w</m> has exactly one solution for every <m>\vec w\in\mathbb R^4</m>.</q>
19+
<p>Here's an example of one such statement:
20+
<!-- {{#statementA}} -->
21+
<q>The vector equation <m>{{veceqleft}}=\vec w</m> has a unique solution for any <m>\vec w\in\mathbb R^4</m>.</q>
22+
<!-- {{/statementA}} -->
23+
<!-- {{#statementB}} -->
24+
<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>
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<!-- {{/statementB}} -->
26+
<!-- {{#statementC}} -->
27+
<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>
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<!-- {{/statementC}} -->
29+
<!-- {{#statementD}} -->
30+
<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>
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<!-- {{/statementD}} -->
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<!-- {{#statementE}} -->
33+
<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>
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<!-- {{/statementE}} -->
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<!-- {{#statementF}} -->
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<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>
37+
<!-- {{/statementF}} -->
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</p>
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</outtro>
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</knowl>

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