strengthen pivotal theorem and deduce uniqueness of rref

src/fields.xml

@@ -120,10 +120,14 @@ the license is included in gfdl.xml.
       where <m>p</m> is a prime number.
       As an example, we can describe a field <m>\F_4</m> with <m>4</m> elements
       <m>0,1,a,a^{-1}</m>, where <m>1+1=0</m> and <m>a^2=a+1</m>.</li>
+      <li>The field of rational functions <m>\Lambda(X)</m> over a field <m>\Lambda</m>
+      consists of rational functions, i.e., those of the form <m>P/Q</m>,
+      where <m>P,Q</m> are polynomials with coefficients from <m>\Lambda</m>
+      and <m>Q\ne 0</m>.</li>
       <li>etc.</li>
     </ul>
   </p>
-  
+
   <p>
   We can now give the definition of abstract vector spaces.</p>
 
@@ -183,20 +187,29 @@ the license is included in gfdl.xml.
     <m>\cdot</m> is called the <term>scalar multiplication</term>; again, it is usually
     left out of notations.
     These operations allow us to produce linear combinations in a meaningful way,
-    and we can prove all the theorems of linear algebra for these.
+    and we can prove all the theorems of linear algebra for these,
+    at least when they are <term>finite dimensional</term>, i.e.,
+    admit a finite ordered basis, as defined in <xref ref="dimension"/>.
     The standard examples are the spaces <m>\Lambda^n</m> of column vectors of
-    length <m>n</m>, for any <m>n\in\N</m>,
-    and vector subspaces (<xref ref="subspaces"/>) of these.
-    Another example is the set of polynomials
-    <me>
-      \Lambda[X] = \{\;a_nX^n + \cdots + a_1X + a_0 \mid a_i\in\Lambda\;\}
-    </me>
-    with the obvious addition and scalar multiplications.
-    The set of <m>m\times n</m> matrices over <m>\Lambda</m>
-    (<xref ref="matrix-equations" />), as well as the set of
-    linear maps <m>f : V \to W</m> between two vector spaces over <m>\Lambda</m>
-    (<xref ref="linear-transformations" />),
-    furnish further examples.
+    length <m>n</m>, for any <m>n\in\N</m>. Other examples:
+    <ul>
+      <li>Vector subspaces (<xref ref="subspaces"/>) of any vector space.</li>
+      <li>The set of polynomials
+      <me>
+        \Lambda[X] = \{\;a_nX^n + \cdots + a_1X + a_0 \mid a_i\in\Lambda\;\}
+      </me>
+      with the obvious addition and scalar multiplications.</li>
+      <li>The set of <m>m\times n</m> matrices over <m>\Lambda</m>
+      (<xref ref="matrix-equations" />).</li>
+      <li>The set of linear maps <m>f : V \to W</m> between two vector spaces
+      over <m>\Lambda</m> (<xref ref="linear-transformations" />).</li>
+      <li>The quotient space <m>V/U</m> of a vector space <m>V</m> with respect
+      to a subspace <m>U</m>: This is obtained a the quotient of <m>V</m> by the
+      equivalence relation <m>\sim</m> defined by <m>x\sim y</m> if and only if
+      <m>y-x \in U</m>. The operations are inhered from <m>V</m> in the sense that
+      <m>\lambda[x] = [\lambda x]</m> and <m>[x]+[y]=[x+y]</m> are well defined.</li>
+      <li>etc.</li>
+    </ul>
   </p>
 
   <p>To read more about abstract algebra, including the theory of commutative rings

src/linindep.xml

@@ -756,11 +756,15 @@ Let's explain why the vectors <m>(1,1,0)</m> and <m>(-2,0,1)</m> are linearly in
       <statement>
         <p>
           Let <m>v_1,v_2,\ldots,v_k</m> be vectors in <m>\R^n</m>, and consider the matrix
-          <me>A = \mat{| |, , |; v_1 v_2 \cdots, v_k; | |, , |}.</me>
+          <me>A = \mat[c]{| |, , |; v_1 v_2 \cdots, v_k; | |, , |}.</me>
           Then we can delete the columns of <m>A</m> <em>without</em> pivots (the columns corresponding to the free variables), without changing <m>\Span\{v_1,v_2,\ldots,v_k\}</m>.
         </p>
 
         <p>The pivot columns are linearly independent, so we cannot delete any more columns without changing the span.</p>
+
+        <p>Every non-pivot column belong to the span of the pivot columns to its left,
+        more precisely, it is the linear combination with coefficients given by the non-pivot column
+        itself in the reduced row echelon form.</p>
       </statement>
 
       <proof>
@@ -799,6 +803,11 @@ Let's explain why the vectors <m>(1,1,0)</m> and <m>(-2,0,1)</m> are linearly in
       </proof>
     </theorem>
 
+    <p>
+      As a corollary, we note the uniqueness of the reduced row echelon form,
+      already stated in <xref ref="row-reduction-works"/>, by uniqueness of the coefficients
+    expressing a non--pivot column as a linear combination of pivot columns to its left.</p>
+
     <p>
       Note that it is necessary to row reduce <m>A</m> to find which are its <xref ref="defn-pivot-pos" text="title">pivot columns</xref>.  However, the span of the columns of the row reduced matrix is generally <em>not</em> equal to the span of the columns of <m>A</m>: one must use the pivot columns of the <em>original</em> matrix.  See <xref ref="dimension-basis-colspace"/> for a restatement of the above theorem.
     </p>
@@ -822,7 +831,7 @@ Let's explain why the vectors <m>(1,1,0)</m> and <m>(-2,0,1)</m> are linearly in
       <title>Pivot Columns and Dimension</title>
       <p>
         Let <m>d</m> be the number of pivot columns in the matrix
-        <me>A = \mat{| |, , |; v_1 v_2 \cdots, v_k; | |, , |}.</me>
+        <me>A = \mat[c]{| |, , |; v_1 v_2 \cdots, v_k; | |, , |}.</me>
         <ul>
           <li>If <m>d=1</m> then <m>\Span\{v_1,v_2,\ldots,v_k\}</m> is a line.</li>
           <li>If <m>d=2</m> then <m>\Span\{v_1,v_2,\ldots,v_k\}</m> is a plane.</li>

src/matrix-coords.xml

@@ -388,7 +388,7 @@ the license is included in gfdl.xml.
     for the domain and the codomain independently.
     In <xref ref="chap-eigenvalues"/> we'll see that the situation is much more subtle
     for linear transformations <m>T : \R^n \to \R^n</m> where the domain and
-    the codomain coincide and we want to use the <em>one</em> basis for both.
+    the codomain coincide and we want to use the <em>same</em> basis for both.
   </p>
 
   <!--

src/row-reduction.xml

@@ -566,7 +566,9 @@ the license is included in gfdl.xml.
     <p>We will give an algorithm, called <term>row reduction</term> or <term>Gaussian elimination</term>, which demonstrates that every matrix is row equivalent to <em>at least one</em> matrix in reduced row echelon form.</p>
 
     <bluebox>
-      <p>The uniqueness statement is interesting<mdash/>it means that, no matter <em>how</em> you row reduce, you <em>always</em> get the same matrix in reduced row echelon form.</p>
+      <p>The uniqueness statement is interesting<mdash/>it means that, no matter <em>how</em> you row reduce,
+      you <em>always</em> get the same matrix in reduced row echelon form.
+      We prove it in <xref ref="linear-independence"/>.</p>
     </bluebox>
 
     <p>This assumes, of course, that you only do the three legal row operations, and you don<rsq/>t make any arithmetic errors.</p>

src/subspace-sum-int.xml

@@ -104,7 +104,7 @@ the license is included in gfdl.xml.
       that is, the columns coming from <m>A</m>.
       (Indeed, we must have <m>\dim(U) \le \dim(U+V)</m>.)
       The non-pivot columns give us a basis for the null space of <m>\mat{A B}</m>.
-      The is because this null space consists of <m>(k+\ell)</m>-dimensional
+      This is because this null space consists of <m>(k+\ell)</m>-dimensional
       vectors <m>\vec{x y}</m> such that <m>Ax + By = 0</m>, or equivalently,
       <m>By = -Ax</m>. So any such pair of coefficient vectors <m>x</m> and <m>y</m>
       gives an element of the intersection <m>U \cap V = \Col(A) \cap \Col(B)</m>.