Emphasize transformed basis vectors

source/linear-algebra/exercises/outcomes/AT/AT2/generator.sage

@@ -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)],
         }
 

source/linear-algebra/exercises/outcomes/AT/AT2/template.xml

@@ -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>