Image descriptions for Linear Algebra

source/linear-algebra/source/02-EV/02.ptx

@@ -50,6 +50,9 @@ since <m>\IR^1=\setBuilder{cx}{c\in\IR}</m>.
     \draw (0,-0.2) -- (0,0.2) node[above] {0};
 \end{tikzpicture}
                     </latex-image>
+                <description>The real line, with a blue vector pointing to the right beginning at <m>0</m> and ending at <m>x</m>,
+                                representing an arbitrary vector in <m>\IR^1</m>.
+                </description>
             </image>
     </figure>
 </observation>
@@ -69,6 +72,8 @@ since <m>\IR^1=\setBuilder{cx}{c\in\IR}</m>.
     \draw[&lt;-&gt;] (0,-4) -- (0,4);
 \end{tikzpicture}
                 </latex-image>
+                <shortdescription>The x-y plane.</shortdescription>
+                <description>The <m>xy</m>-plane.</description>.
             </image>
         </figure>
   <ol marker="A." cols="2">
@@ -116,6 +121,8 @@ since <m>\IR^1=\setBuilder{cx}{c\in\IR}</m>.
     \draw[-&gt;] (0,0,0) -- (0,0,6);
   \end{tikzpicture}
             </latex-image>
+            <shortdescription>The coordinate axes representing three dimensional space.</shortdescription>
+            <description>Coordinate axes representing <m>\IR^3</m>.</description>
         </image>
         </figure>
   <ol marker="A." cols="2">
@@ -173,6 +180,10 @@ since <m>\IR^1=\setBuilder{cx}{c\in\IR}</m>.
     \draw[thick,red,-&gt;] (0,0,0) -- (-2,0,1);
   \end{tikzpicture}
             </latex-image>
+        <description><p>Two images side by side. The left image is the <m>xy</m>-plane, with a single vector pointing down and to the right.
+                    Its spanning set, the line parallel to it, is also shown.</p>
+                    <p>The right image illustrates two non-parallel vectors in three dimensional space, as well as the two-dimensional plane they span.</p>
+        </description>
         </image>
         </figure>
     </statement>

source/linear-algebra/source/02-EV/04.ptx

@@ -97,6 +97,7 @@
     \draw[thick,purple,->] (0,0,0) -- (1,1,-1);
   \end{tikzpicture}
 				</latex-image>
+                <description>Three vectors in <m>\IR^3</m>, no two of which are parallel. They all lie in the same plane, which is shown.</description>
 			</image>
         </figure>
         <p>

source/linear-algebra/source/03-AT/01.ptx

@@ -105,6 +105,11 @@ Given a linear transformation <m>T:V\to W</m>,
   \end{scope}
 \end{tikzpicture}
 			</latex-image>
+      <description>
+        The domain <m>\IR^3</m> is represented on the left by the <m>xyz</m> coordinate axes, along with an arbitrary vector <m>\vec{v}</m>.
+        A curved dotted arrow to the right points to the co-domain, <m>\IR^2</m>, represented by the <m>xy</m> coordinate axes, along with
+        an arbitrary vector labeled <m>T(\vec{v})</m>.
+      </description>
 		</image>
     </figure>
     </statement>
@@ -118,7 +123,13 @@ as is necessary for computer animation in film or video games.
         </p>
     <figure xml:id="figure-teapot-projection">
     <caption>A projection of a <m>3D</m> teapot onto a <m>2D</m> screen</caption>
-		<image source="teapot.png" xml:id="AT1-image-teapot"/>
+		<image source="teapot.png" xml:id="AT1-image-teapot">
+      <description>
+        <p>A computer generated image of a three dimensional teapot sitting in front of a screen that shows a 
+        flattened, two dimensional image of the same teapot.  Several parallel black arrows point from identifiable points on the three 
+        dimensional teapot (such as the spout and handle) to the corresponding places on the two dimensional image.</p>
+      </description>
+    </image>
     </figure>
 </observation>
 

source/linear-algebra/source/03-AT/03.ptx

@@ -110,6 +110,12 @@ is an important subspace of <m>V</m> defined by
     \end{scope}
   \end{tikzpicture}
 					</latex-image>
+          <description>Two Euclidean spaces are shown, connected by a curved dashed arrow to the right.
+            The space on the left is <m>\IR^3</m>, represented by the <m>xyz</m> coordinate axes.  A blue
+            line representing a one dimensional subspace is shown and labeled <m>\ker T</m>.  The space on the 
+            right is <m>\IR^2</m>, represented by the <m>xy</m>-plane. The zero vector <m>\vec{0}</m> is shown
+            and labeled.
+          </description>
 				</image>
             </figure>
     </statement>
@@ -307,6 +313,17 @@ right example's image is a planar subspace of <m>\IR^3</m>.
   \end{scope}
 \end{tikzpicture}
 				</latex-image>
+        <description>
+          <p>Two examples are shown.  The one on the left illustrates a transformation from <m>\IR^3</m> to <m>\IR^2</m>
+          with the <m>xyz</m> coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed
+          arrow to the right to an image of the <m>xy</m>-plane.  The images of the individual arbitrary vectors are shown,
+          and the entire <m>xy</m> plane is shaded, representing that they span the entire space <m>\IR^2</m>.</p>
+          <p>The example on the right illustrates a transformation from <m>\IR^2</m>to <m>\IR^3</m>
+          with the <m>xy</m> coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed
+          arrow to the right to an image of the <m>xyz</m> coordinate axes.  The images of the individual arbitrary vectors are 
+          shown, all lying in a two dimensional plane which is shaded.
+          </p>
+        </description>
 			</image>
         </figure>
     </statement>

source/linear-algebra/source/03-AT/04.ptx

@@ -88,6 +88,18 @@ distinct vectors to the same place.  More precisely, <m>T</m> is injective if
   \node[anchor=north] at (5,-1) {not injective};
 \end{tikzpicture}
 				</latex-image>
+        <description>
+          <p>Two examples are shown.  The one on the left illustrates a transformation from <m>\IR^2</m> to <m>\IR^3</m>
+          with the <m>xy</m> coordinate axes (with two distinct vectors labeled <m>\vec{v}</m> and <m>\vec{w}</m> shown) 
+          connected by a curved dashed arrow to the right to an image of the <m>xyz</m> coordinate axes.  The images of 
+          the individual vectors are shown and are distinct, labeled <m>T(\vec{v})</m> and <m>T(\vec{w})</m>. This example
+          is labeled below as "injective".</p>
+          <p>The example on the right illustrates a transformation from <m>\IR^3</m>to <m>\IR^3</m>
+          with the <m>xyz</m> coordinate axes (with two distinct vectors labeled <m>\vec{v}</m> and <m>\vec{w}</m> shown) 
+          connected by a curved dashed arrow to the right to an image of the <m>xy</m> coordinate axes.  The images of 
+          the individual vectors are shown and are the same, labeled <m>T(\vec{v})=T(\vec{w})</m>. This example
+          is labeled below as "not injective".</p>
+        </description>
 			</image>
         </figure>
     </statement>
@@ -252,6 +264,18 @@ Let <m>T: V \rightarrow W</m> be a linear transformation.
   \node[anchor=north] at (5,-2) {not surjective};
 \end{tikzpicture}
 			</latex-image>
+        <description>
+          <p>Two examples are shown.  The one on the left illustrates a transformation from <m>\IR^3</m> to <m>\IR^2</m>
+          with the <m>xyz</m> coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed
+          arrow to the right to an image of the <m>xy</m>-plane.  The images of the individual arbitrary vectors are shown,
+          and the entire <m>xy</m> plane is shaded, representing that they span the entire space <m>\IR^2</m>.  This example is
+          labeled below as "surjective".</p>
+          <p>The example on the right illustrates a transformation from <m>\IR^2</m>to <m>\IR^3</m>
+          with the <m>xy</m> coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed
+          arrow to the right to an image of the <m>xyz</m> coordinate axes.  The images of the individual arbitrary vectors are 
+          shown, all lying in a two dimensional plane which is shaded.  This example is labeled below as "not surjective".
+          </p>
+        </description>
 		</image>
     </figure>
     </statement>
@@ -435,6 +459,12 @@ recognized by its <term>trivial</term> kernel.
   \end{scope}
 \end{tikzpicture}
 				</latex-image>
+        <description>
+          The <m>xy</m>-plane on the left, with the zero vector <m>\vec{0}</m> and two distinct, non-parallel arbitrary vectors
+          labeled <m>\vec{v}</m> and <m>\vec{w}</m>, connected by a curved dashed arrow to the right to the <m>xyz</m> coordinate axes,
+          representing <m>\IR^3</m>.  Three vectors are shown in <m>\IR^3</m>: the zero vector, labeled <m>T(\vec{0})=\vec{0}</m>; 
+          and nonzero, non-parallel vectors <m>T(\vec{v})</m> and <m>T(\vec{w})</m>.
+        </description>
 			</image>
         </figure>
     </statement>
@@ -537,6 +567,19 @@ recognized by its identical codomain and image.
   \node[anchor=north] at (5,-2) {not surjective, \(\Im T\not=\IR^3\)};
 \end{tikzpicture}
 			</latex-image>
+        <description>
+          <p>Two examples are shown.  The one on the left illustrates a transformation from <m>\IR^3</m> to <m>\IR^2</m>
+          with the <m>xyz</m> coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed
+          arrow to the right to an image of the <m>xy</m>-plane.  The images of the individual arbitrary vectors are shown,
+          and the entire <m>xy</m> plane is shaded, representing that they span the entire space <m>\IR^2</m>.  This example is
+          labeled below as "surjective, <m>\Im T = \IR^2</m>".</p>
+          <p>The example on the right illustrates a transformation from <m>\IR^2</m>to <m>\IR^3</m>
+          with the <m>xy</m> coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed
+          arrow to the right to an image of the <m>xyz</m> coordinate axes.  The images of the individual arbitrary vectors are 
+          shown, all lying in a two dimensional plane which is shaded.  This example is labeled below as "not surjective, 
+          <m>\Im T \neq \IR^3</m>".
+          </p>
+        </description>
 		</image>
     </figure>
     </statement>
@@ -756,6 +799,18 @@ dimension than its codomain, and is therefore not surjective.
       \node[anchor=north] at (5,-2) {not surjective, \(2&lt;3\)};
     \end{tikzpicture}
 			</latex-image>
+        <description>
+          <p>Two examples are shown.  The one on the left illustrates a transformation from <m>\IR^3</m> to <m>\IR^2</m>
+          with the <m>xyz</m> coordinate axes (with two non-parallel vectors <m>\vec{v}</m> and <m>\vec{w}</m> shown) connected by a curved dashed
+          arrow to the right to an image of the <m>xy</m>-plane.  A single vector is shown in the <m>xy</m>-plane, labeled
+          <m>T(\vec{v})=T(\vec{w})</m>.  This example is labeled below as "not injective, <m>3&gt;2</m>".</p>
+          <p>The example on the right illustrates a transformation from <m>\IR^2</m>to <m>\IR^3</m>
+          with the <m>xy</m> coordinate axes (with several arbitrary vectors illustrated) connected by a curved dashed
+          arrow to the right to an image of the <m>xyz</m> coordinate axes.  The images of the individual arbitrary vectors are 
+          shown, all lying in a two dimensional plane which is shaded.  This example is labeled below as "not surjective, 
+          <m>2 &lt;3</m>".
+          </p>
+        </description>
 		</image>
     </figure>
     <p>

source/linear-algebra/source/04-MX/01.ptx

@@ -50,6 +50,12 @@ is a linear map from <m>\IR^n \rightarrow \IR^k</m>.
 \IR^n \arrow[rr, bend right, "S\circ T"']  \arrow[r,"T"] \&amp; \IR^m  \arrow[r,"S"]  \&amp;\IR^k 
 \end{tikzcd}
 				</latex-image>
+                <description>
+                    A representation of the composition of maps.  The chain <m>\IR^n \rightarrow \IR^m \rightarrow \IR^k</m> is
+                    adorned with a <m>T</m> labeling the arrow from <m>\IR^n</m> to <m>\IR^m</m>, and a <m>S</m> labeling the arrow
+                    from <m>\IR^m \rightarrow \IR^k</m>.  Below this is a curved arrow connecting <m>\IR^n</m> on the left to <m>\IR^k</m>
+                    on the right, which is labeled <m>S \circ T</m>.
+                </description>
 			</image>
 			<caption>The composition of two linear maps.</caption>
         </figure>

source/linear-algebra/source/05-GT/01.ptx

@@ -76,6 +76,14 @@ transforms the unit square.
 \draw[red,dashed] (1,0) -- (1,1) -- (0,1);
 \end{tikzpicture}
 		</latex-image>
+    <description>
+      Two vectors are shown in the <m>xy</m>-plane: 
+      A blue vector a long the <m>x</m>-axis is labeled <m>A\vec{e}_1 =\left[\begin{array}{c}2 \\ 0 \end{array}\right]</m> and 
+      a blue vector a long the <m>y</m>-axis is labeled <m>A\vec{e}_2 =\left[\begin{array}{c}0 \\ 3 \end{array}\right]</m>.
+      Dashed blue lines extend vertically and horizontally from the ends of these vectors to illustrate a rectangle formed
+      with those two vectors as two of the sides.  Additionally, a shaded red unit square is similarly illustrated by the
+      red vectors <m>\left[\begin{array}{c}1 \\ 0\end{array}\right]</m> and <m>\left[\begin{array}{c}0 \\ 1\end{array}\right]</m>. 
+    </description>
 	</image>
 	<caption>Transformation of the unit square by the matrix <m>A</m>.</caption>
 </figure>
@@ -97,7 +105,7 @@ The image below illustrates how the linear transformation
 standard matrix <m>B = \left[\begin{array}{cc} 2 &amp; 3 \\ 0 &amp; 4 \end{array}\right]</m>
 transforms the unit square.
         </p>
-<figure>
+<figure xml:id="fig-GT1-image-unit-square-transform2">
 	<image width="75%" xml:id="GT1-image-unit-square-transform2">
 		<latex-image>
 \begin{tikzpicture}
@@ -112,6 +120,14 @@ transforms the unit square.
 \draw[red,dashed] (1,0) -- (1,1) -- (0,1);
 \end{tikzpicture}
 		</latex-image>
+    <description>
+      Two vectors are shown in the <m>xy</m>-plane: 
+      A blue vector a long the <m>x</m>-axis is labeled <m>B\vec{e}_1 =\left[\begin{array}{c}2 \\ 0 \end{array}\right]</m> and 
+      a blue vector extending upwards to the right is labeled <m>B\vec{e}_2 =\left[\begin{array}{c}3 \\ 4 \end{array}\right]</m>.
+      Dashed blue lines extend from the ends of these vectors parallel to the other vector to illustrate a parallelogram formed
+      with those two vectors as two of the sides.  Additionally, a shaded red unit square is similarly illustrated by the
+      red vectors <m>\left[\begin{array}{c}1 \\ 0\end{array}\right]</m> and <m>\left[\begin{array}{c}0 \\ 1\end{array}\right]</m>. 
+    </description>
 	</image>
 	<caption>Transformation of the unit square by the matrix <m>B</m></caption>
 </figure>
@@ -146,7 +162,7 @@ What is the area of the transformed unit square?
       =
     4\left[\begin{array}{c}\frac{3}{4}\\\frac{1}{2}\end{array}\right]
   </me>
-  <figure>
+  <figure xml:id="fig-GT1-image-scale-not-rotate">
 	<image width="75%" xml:id="GT1-image-scale-not-rotate">
 		<latex-image>
   \begin{tikzpicture}
@@ -161,6 +177,14 @@ What is the area of the transformed unit square?
   \draw[blue,dashed] (2,0) -- (5,2) -- (3,2);
   \end{tikzpicture}
 		</latex-image>
+    <description>
+      Two vectors are shown in the <m>xy</m>-plane: 
+      A blue vector a long the <m>x</m>-axis is labeled <m>B\left[\begin{array}{c}1 \\ 0 \end{array}\right] =2\left[\begin{array}{c}1 \\ 0 \end{array}\right]</m> and 
+      a blue vector extending to the right and upwards is labeled <m>B\left[\begin{array}{c}\frac{3}{4} \\ \frac{1}{2} \end{array}\right] =4\left[\begin{array}{c}\frac{3}{4} \\ \frac{1}{2} \end{array}\right]</m>.
+      Dashed blue lines extend from the ends of these vectors parallel to the other vector to illustrate a parallelogram formed
+      with those two vectors as two of the sides.  Additionally, a shaded red parallelogram is similarly illustrated by the
+      red vectors <m>\left[\begin{array}{c}1 \\ 0\end{array}\right]</m> and <m>\left[\begin{array}{c}\frac{3}{4} \\ \frac{1}{2}\end{array}\right]</m>. 
+    </description>
 	</image>
 	<caption>Certain vectors are stretched out without being rotated.</caption>
   </figure>
@@ -206,6 +230,9 @@ What is the area of the transformed unit square?
   \draw[blue,dashed] (2,0) -- (5,2) -- (3,2);
   \end{tikzpicture}
 		</latex-image>
+  <description>
+    The images from <xref ref="fig-GT1-image-unit-square-transform2"/> and <xref ref="fig-GT1-image-scale-not-rotate"/> are shown side by side.
+  </description>
 	</image>
 	<caption>A linear map transforming parallelograms into parallelograms.</caption>
 </figure>
@@ -250,6 +277,9 @@ In order to figure out how to compute it, we first figure out the properties it
   \draw[blue,dashed] (2,0) -- (5,2) -- (3,2);
   \end{tikzpicture}
 		</latex-image>
+  <description>
+    The images from <xref ref="fig-GT1-image-unit-square-transform2"/> and <xref ref="fig-GT1-image-scale-not-rotate"/> are shown side by side.
+  </description>
 	</image>
 	<caption>The linear transformation <m>B</m> scaling areas by a constant factor, which we call the <term>determinant</term></caption>
 </figure>
@@ -277,7 +307,13 @@ area of resulting parallelogram, what is the value of <m>\det([\vec{e}_1\hspace{
 \draw[dashed,blue] (1,0) -- (1,1);
 \draw[dashed,blue] (0,1) -- (1,1);
 \end{tikzpicture}
-		</latex-image>
+		</latex-image>    <description>
+      Two vectors are shown in the <m>xy</m>-plane: 
+      A blue vector a long the <m>x</m>-axis is labeled <m>\vec{e}_1=\left[\begin{array}{c}1 \\ 0 \end{array}\right]</m> and 
+      a blue vector extending along the <m>y</m>-axis is labeled <m>\vec{e}_2 =\left[\begin{array}{c}0 \\ 1 \end{array}\right]</m>.
+      Dashed blue lines extend from the ends of these vectors parallel to the other vector to illustrate a square formed
+      with those two vectors as two of the sides.  This square is shaded red. 
+    </description>
 	</image>
 	<caption>The transformation of the unit square by the identity matrix.</caption>
 </figure>
@@ -485,6 +521,14 @@ the same parallelogram, but the second matrix reflects its orientation.
 \draw[red,dashed] (1,0) -- (1,1) -- (0,1);
 \end{tikzpicture}
 		</latex-image>
+    <description>
+      Two images are shown.  Both show the <m>xy</m>-plane along with two blue vectors (one extending right, and one extending
+      upwards and to the right), with dotted lines completing the parallelogram they span.  Both images also contain the red unit
+      square.  The image on the left labels the blue vectors as 
+      <m>A\vec{e}_1=\left[\begin{array}{c}2 \\ 0 \end{array}\right]</m> and <m>A\vec{e}_2=\left[\begin{array}{c}3 \\ 4 \end{array}\right]</m>,
+      while the image on the right labels the two vectors as
+      <m>B\vec{e}_2=\left[\begin{array}{c}2 \\ 0 \end{array}\right]</m> and <m>B\vec{e}_1=\left[\begin{array}{c}3 \\ 4 \end{array}\right]</m>.
+    </description>
 	</image>
 	<caption>Reflection of a parallelogram as a result of swapping columns.</caption>
 </figure>
@@ -584,6 +628,11 @@ may be verified by adding and subtracting columns.
 \end{scope}
 \end{tikzpicture}
 		</latex-image>
+    <description>
+      A red square, a larger purple parallelogram, and an even larger blue rectangle are arranged from left to right horizontally.
+      A curved arrow labeled <m>B</m> points from the red square to the purple parallelogram, and a curved arrow labeled <m>A</m>
+      points from the purple parallelogram to the blue rectangle.
+    </description>
 	</image>
 	<caption>Area changing under the composition of two linear maps</caption>
 </figure>

source/linear-algebra/source/05-GT/02.ptx

@@ -79,6 +79,16 @@ into the determinant of a smaller matrix.
   \draw[purple,dashed,very thick] (0,0,0) -- node[left] {\tiny\(h=1\)} (0,1,0);
   \end{tikzpicture}
 		</latex-image>
+    <description>
+      Three vectors in <m>\IR^3</m> are shown in red: 
+      <m>\left[\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right]</m>,
+      <m>\left[\begin{array}{c}1 \\ 3 \\ 0 \end{array}\right]</m>, and
+      <m>\left[\begin{array}{c}0 \\ 1 \\ 1 \end{array}\right]</m>.
+      The parallelogram formed by <m>\left[\begin{array}{c}1 \\ 1 \\ 0 \end{array}\right]</m> and 
+      <m>\left[\begin{array}{c}1 \\ 3 \\ 0 \end{array}\right]</m>, lying in the <m>xy</m>-plane, is shaded red.
+      The parallelepiped formed by the three vectors is shown in blue.
+      Additionally, a dashed red line segment along the <m>z</m> axis spanning the apparent height of the parallelepiped is labeled <m>h=1</m>.
+    </description>
 	</image>
 	<caption>Transformation of the unit cube by the linear transformation.</caption>
   </figure>