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src/sectionLI.xml
<p><acroref type="theorem" acro="LIVHS" /> suggests we place these vectors into a matrix as columns and analyze the rowreduced version of the matrix,
+<p><acroref type="theorem" acro="LIVRN" /> suggests we place these vectors into a matrix as columns and analyze the rowreduced version of the matrix,
to recognize, via <acroref type="theorem" acro="LIVHS" /> that $S$ is a linearly dependent set. Boom!</p>
+to recognize, via <acroref type="theorem" acro="LIVRN" /> that $S$ is a linearly dependent set. Boom!</p>
<p>To employ <acroref type="theorem" acro="LIVHS" />, we form a $4\times 9$coefficientmatrix, $C$,
+<p>To employ <acroref type="theorem" acro="LIVHS" />, we form a $4\times 9$ matrix, $C$, whose columns are the vectors in $B$
<p>To determine if the homogeneous system $\linearsystem{C}{\zerovector}$ has a unique solution or not, we would normally rowreduce this matrix. But in this particular example, we can do better. <acroref type="theorem" acro="HMVEI" /> tells us that since the system is homogeneous with $n=9$ variables in $m=4$ equations, and $n>m$, there must be infinitely many solutions. Since there is not a unique solution, <acroref type="theorem" acro="LIVHS" /> says the set is linearly dependent.
+<p>To determine if the homogeneous system $\homosystem{C}$ has a unique solution or not, we would normally rowreduce this matrix. But in this particular example, we can do better. <acroref type="theorem" acro="HMVEI" /> tells us that since the system is homogeneous with $n=9$ variables in $m=4$ equations, and $n>m$, there must be infinitely many solutions. Since there is not a unique solution, <acroref type="theorem" acro="LIVHS" /> says the set is linearly dependent.
<p>The situation in <acroref type="example" acro="LLDS" /> is slick enough to warrant formulating as a theorem.
+<p>Suppose that $S=\set{\vectorlist{u}{n}}\subseteq\complex{m}$ and $n>m$. Then $S$ is a linearly dependent set.</p>
<p>Form the $m\times n$ coefficient matrix $A$ that has the column vectors $\vect{u}_i$, $1\leq i\leq n$ as its columns. Consider the homogeneous system $\homosystem{A}$. By <acroref type="theorem" acro="HMVEI" /> this system has infinitely many solutions. Since the system does not have a unique solution, <acroref type="theorem" acro="LIVHS" /> says the columns of $A$ form a linearly dependent set, which is the desired conclusion.
+<p>Form the $m\times n$ matrix $A$ whose columns are $\vect{u}_i$, $1\leq i\leq n$. Consider the homogeneous system $\homosystem{A}$. By <acroref type="theorem" acro="HMVEI" /> this system has infinitely many solutions. Since the system does not have a unique solution, <acroref type="theorem" acro="LIVHS" /> says the columns of $A$ form a linearly dependent set, as desired.