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+1 1changes.txt

+25 32src/sectionNM.xml
src/sectionNM.xml
<p>Notice that an identity matrix is square, and in reduced rowechelon form. So in particular, if we were to arrive at the identity matrix while bringing a matrix to reduced rowechelon form, then it would have all of the diagonal entries circled as leading 1's.</p>
+<p>Notice that an identity matrix is square, and in reduced rowechelon form. Also, every column is a pivot column, and every possible pivot column appears once.</p>
+<p>We have the coefficient matrix for <acroref type="archetype" acro="A" /> and a rowequivalent matrix $B$ in reduced rowechelon form,
+A=<archetypepart acro="A" part="purematrix" />\rref<archetypepart acro="A" part="matrixreduced" />=B</equation>
<p>Sincethis matrixis not the $3\times 3$ identity matrix, <acroref type="theorem" acro="NMRRI" /> tells us that $A$ is a singular matrix.</p>
+<p>Since $B$ is not the $3\times 3$ identity matrix, <acroref type="theorem" acro="NMRRI" /> tells us that $A$ is a singular matrix.</p>
+<p>We have the coefficient matrix for <acroref type="archetype" acro="B" /> and a rowequivalent matrix $B$ in reduced rowechelon form,
+A=<archetypepart acro="A" part="purematrix" />\rref<archetypepart acro="B" part="matrixreduced" />=B</equation>
<p>Sincethis matrixis the $3\times 3$ identity matrix, <acroref type="theorem" acro="NMRRI" /> tells us that $A$ is a nonsingular matrix.</p>
+<p>Since $B$ is the $3\times 3$ identity matrix, <acroref type="theorem" acro="NMRRI" /> tells us that $A$ is a nonsingular matrix.</p>
the null space is the set of solutions to the homogeneous system of equations $\homosystem{A}$ has a solution set and null space constructed in <acroref type="example" acro="HISAA" /> as
+<p>Given the singular coefficient matrix from <acroref type="archetype" acro="A" />, the null space is the set of solutions to the homogeneous system of equations $\homosystem{A}$ has a solution set and null space constructed in <acroref type="example" acro="HISAA" /> as an infinite set of vectors.
+A<![CDATA[&=]]><archetypepart acro="A" part="purematrix" /><![CDATA[&]]>\nsp{A}<![CDATA[&=]]>\setparts{\colvector{x_3\\x_3\\x_3}}{x_3\in\complex{\null}}
the solution set to the homogeneous system $\homosystem{A}$ is constructed in <acroref type="example" acro="HUSAB" /> and contains only the trivial solution, so the null space has only a single element,
+<p>Given the nonsingular coefficient matrix from <acroref type="archetype" acro="B" />, the solution set to the homogeneous system $\homosystem{A}$ is constructed in <acroref type="example" acro="HUSAB" /> and contains only the trivial solution, so the null space of $A$ has only a single element,
+A<![CDATA[&=]]><archetypepart acro="B" part="purematrix" /><![CDATA[&]]>\nsp{A}<![CDATA[&=]]>\set{\colvector{0\\0\\0}}
<p>Suppose that $A$ is a square matrix. Then $A$ is nonsingular if and only if the null space of $A$, $\nsp{A}$, contains only the zero vector, <ie /> $\nsp{A}=\set{\zerovector}$.</p>
+<p>Suppose that $A$ is a square matrix. Then $A$ is nonsingular if and only if the null space of $A$ is the set containing only the zero vector, <ie /> $\nsp{A}=\set{\zerovector}$.</p>
+<p>The statement that $A$ is nonsingular is equivalent to each of the subsequent statements by, in turn,
<acroref type="theorem" acro="NMUS" />. So the statement of this theorem is just a convenient way to organize all these results.</p>
<p>Finally, you may have wondered why we refer to a matrix as <em>nonsingular</em> when it creates systems of equations with <em>single</em> solutions (<acroref type="theorem" acro="NMUS" />)! I've wondered the same thing. We'll have an opportunity to address this when we get to <acroref type="theorem" acro="SMZD" />. Can you wait that long?</p>
+<p>Finally, you may have wondered why we refer to a matrix as <em>nonsingular</em> when it creates systems of equations with <em>single</em> solutions (<acroref type="theorem" acro="NMUS" />)! I've wondered the same thing. We will have an opportunity to address this when we get to <acroref type="theorem" acro="SMZD" />. Can you wait that long?</p>
<li>Suppose we have a system of equations and its coefficient matrix is nonsingular. What can you say about the solution set for this system?