DOC: change notation, add intermediate logic

Change the notation of the row, column indices of the matrix.

Add intermediate step for showing why inverse of affine is mapping from
mm to voxels.

Author: matthew-brett

doc/source/coordinate_systems.rst

@@ -515,16 +515,16 @@ We could record the parameters necessary for $f$ as the 3 by 3 matrix, $M$
 and the 3 by 1 vector $(a, b, c)$.
 
 In fact, the 4 by 4 image *affine array* does include exactly this
-information. If $m_{ij}$ is the value in row $i$ column $j$ of matrix $M$,
+information. If $m_{i,j}$ is the value in row $i$ column $j$ of matrix $M$,
 then the image affine matrix $A$ is:
 
 .. math::
 
     A =
     \begin{bmatrix}
-    m_{11} & m_{12} & m_{13} & a \\
-    m_{21} & m_{22} & m_{23} & b \\
-    m_{31} & m_{32} & m_{33} & c \\
+    m_{1,1} & m_{1,2} & m_{1,3} & a \\
+    m_{2,1} & m_{2,2} & m_{2,3} & b \\
+    m_{3,1} & m_{3,2} & m_{3,3} & c \\
     0 & 0 & 0 & 1 \\
     \end{bmatrix}
 
@@ -546,9 +546,9 @@ vectors:
     1\\
     \end{bmatrix} =
     \begin{bmatrix}
-    m_{11} & m_{12} & m_{13} & a \\
-    m_{21} & m_{22} & m_{23} & b \\
-    m_{31} & m_{32} & m_{33} & c \\
+    m_{1,1} & m_{1,2} & m_{1,3} & a \\
+    m_{2,1} & m_{2,2} & m_{2,3} & b \\
+    m_{3,1} & m_{3,2} & m_{3,3} & c \\
     0 & 0 & 0 & 1 \\
     \end{bmatrix}
     \begin{bmatrix}
@@ -614,6 +614,19 @@ matrix.  Put another way:
     1\\
     \end{bmatrix}
 
+    A^{-1}\begin{bmatrix}
+    x\\
+    y\\
+    z\\
+    1\\
+    \end{bmatrix} = A^{-1} A
+    \begin{bmatrix}
+    i\\
+    j\\
+    k\\
+    1\\
+    \end{bmatrix}
+
     \begin{bmatrix}
     i\\
     j\\