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Emphasize transformed basis vectors #724

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Original file line number Diff line number Diff line change
@@ -35,10 +35,12 @@ class Generator(BaseGenerator):
"varmap": A*xs,
"varvector": xs,
"Sstandardmatrix": A,
"Scolumns": [{"i": i+1, "col": column_matrix(A.column(i))} for i in range(Scolumns)],
"Trows": Trows,
"Tcols": Tcolumns,
"Tstandardmatrix": B,
"vector": v,
"Tvector": B*v
"Tvector": B*v,
"Tcolumns": [{"i": i+1, "col": column_matrix(B.column(i))} for i in range(Tcolumns)],
}

87 changes: 66 additions & 21 deletions source/linear-algebra/exercises/outcomes/AT/AT2/template.xml
Original file line number Diff line number Diff line change
@@ -1,28 +1,73 @@
<?xml version='1.0' encoding='UTF-8'?>
<knowl mode="exercise" xmlns="https://spatext.clontz.org" version="0.2">
<knowl>
<content>
<p>Explain and demonstrate how to compute
the standard matrix for the linear transformation
<m>S:\mathbb{R}^{{Scols}} \to \mathbb{R}^{{Srows}}</m> given by
<me>S\left( {{varvector}} \right) = {{varmap}}</me>
by computing transformations of the standard basic vectors.</p>
</content>
<outtro>
<p><me>{{Sstandardmatrix}}</me></p>
</outtro>
<intro>
<p>
Consider the linear transformation
<m>S:\mathbb{R}^{{Scols}} \to \mathbb{R}^{{Srows}}</m> given by
<me>S\left( {{varvector}} \right) = {{varmap}}.</me>
</p>
</intro>
<knowl>
<content>
<p>
Compute the transformation of each vector from the standard
basis for <m>\mathbb R^{{Scols}}</m>.
</p>
</content>
<outtro>
<p><me>\hspace{1em}
<!-- {{#Scolumns}} -->
S(\vec e_{ {{i}} }) = {{col}} \hspace{1em}
<!-- {{/Scolumns}} -->
</me></p>
</outtro>
</knowl>
<knowl>
<content>
<p>
Explain and demonstrate how these are used to
form the standard matrix for this transformation.
</p>
</content>
<outtro>
<p><me>{{Sstandardmatrix}}</me></p>
</outtro>
</knowl>
</knowl>
<knowl>
<content>
<p>Let <m>T:\mathbb{R}^{{Tcols}} \to \mathbb{R}^{{Trows}}</m>
be the linear transformation given by the standard matrix
<me>{{Tstandardmatrix}}.</me>
Explain and demonstrate how to compute
<m>T\left({{vector}}\right)</m> by using the values of
transformed standard basic vectors.</p>
</content>
<outtro>
<p><me>T\left({{vector}}\right)={{Tvector}}</me></p>
</outtro>
<intro>
<p>
Consider the linear transformation
<m>T:\mathbb{R}^{{Tcols}} \to \mathbb{R}^{{Trows}}</m>
defined by the standard matrix <me>{{Tstandardmatrix}}.</me>
</p>
</intro>
<knowl>
<content>
<p>
Describe how this map transforms each of the vectors for the
standard basis of <m>\mathbb R^{{Tcols}}</m>.
</p>
</content>
<outtro>
<p><me>\hspace{1em}
<!-- {{#Tcolumns}} -->
T(\vec e_{ {{i}} }) = {{col}} \hspace{1em}
<!-- {{/Tcolumns}} -->
</me></p>
</outtro>
</knowl>
<knowl>
<content>
<p>
Explain and demonstrate how to use these to compute
<m>T\left({{vector}}\right)</m>.
</p>
</content>
<outtro>
<p><me>T\left({{vector}}\right)={{Tvector}}</me></p>
</outtro>
</knowl>
</knowl>
</knowl>