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Minor edits, Section RREF

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# File changes.txt

`+v3.xx 2013//`
`+~~~~~~~~~~~~~~~~`
`+Edit: Minor edits in Sections WILA, SSLE, RREF`
`+`
`+`
` v3.10 2013/08/20`
` ~~~~~~~~~~~~~~~~`
` New:  Exercise MM.T12, Theorem HMIP reprised`

# File src/section-RREF.xml

` `
` `
` </sageadvice>`
`-<p>When we do equation operations on system of equations, the names of the variables really aren't very important.  \$x_1\$, \$x_2\$, \$x_3\$, or \$a\$, \$b\$, \$c\$, or \$x\$, \$y\$, \$z\$, it really doesn't matter.  In this subsection we will describe some notation that will make it easier to describe linear systems, solve the systems and describe the solution sets.  Here is a list of definitions, laden with notation.</p>`
`+<p>When we do equation operations on system of equations, the names of the variables really are not very important.  Use \$x_1\$, \$x_2\$, \$x_3\$, or \$a\$, \$b\$, \$c\$, or \$x\$, \$y\$, \$z\$, it really doesn't matter.  In this subsection we will describe some notation that will make it easier to describe linear systems, solve the systems and describe the solution sets.  Here is a list of definitions, laden with notation.</p>`
` `
` <definition acro="CV" index="vector!column">`
` <title>Column Vector</title>`
` </proof>`
` </theorem>`
` `
`-<p>We will now run through some examples of using these definitions and theorems to solve some systems of equations.  From now on, when we have a matrix in reduced row-echelon form, we will mark the leading 1's with a small box.  In your work, you can box 'em, circle 'em or write 'em in a different color <mdash /> just identify 'em somehow.  This device will prove very useful later and is a <em>very good habit</em> to start developing right now.</p>`
`+   <p>We will now run through some examples of using these definitions and theorems to solve some systems of equations.  From now on, when we have a matrix in reduced row-echelon form, we will mark the leading 1's with a small box.  In your work, you can box 'em, circle 'em or write 'em in a different color <mdash /> just identify 'em somehow.  This device will prove very useful later and is a <em>very good habit</em> to start developing <em>right now</em>.</p>`
` `
` <example acro="SAB" index="archetype B!solutions">`
` <title>Solutions for Archetype B</title>`
` <![CDATA[x_2-x_3&=2.]]>`
` </alignmath></p>`
` `
`-<p>While this system is fairly easy to solve, it also appears to have a multitude of solutions.  For example, choose \$x_3=1\$ and see that then \$x_1=2\$ and \$x_2=3\$ will together form a solution.  Or choose \$x_3=0\$, and then discover that \$x_1=3\$ and \$x_2=2\$ lead to a solution.  Try it yourself: pick <em>any</em> value of \$x_3\$ you please, and figure out what \$x_1\$ and \$x_2\$ should be to make the first and second equations (respectively) true.  We'll wait while you do that.  Because of this behavior, we say that \$x_3\$ is a <q>free</q> or <q>independent</q> variable.  But why do we vary \$x_3\$ and not some other variable?  For now, notice that the third column of the augmented matrix does not have any leading 1's in its column.  With this idea, we can rearrange the two equations, solving each for the variable that corresponds to the leading 1 in that row.`
`+<p>While this system is fairly easy to solve, it also appears to have a multitude of solutions.  For example, choose \$x_3=1\$ and see that then \$x_1=2\$ and \$x_2=3\$ will together form a solution.  Or choose \$x_3=0\$, and then discover that \$x_1=3\$ and \$x_2=2\$ lead to a solution.  Try it yourself: pick <em>any</em> value of \$x_3\$ you please, and figure out what \$x_1\$ and \$x_2\$ should be to make the first and second equations (respectively) true.  We'll wait while you do that.  Because of this behavior, we say that \$x_3\$ is a <q>free</q> or <q>independent</q> variable.  But why do we vary \$x_3\$ and not some other variable?  For now, notice that the third column of the augmented matrix does not have any leading 1's in its column and so is not a pivot column.  With this idea, we can rearrange the two equations, solving each for the variable that corresponds to the leading 1 in that row.`
` <alignmath>`
` <![CDATA[x_1&=3-x_3\\]]>`
` <![CDATA[x_2&=2+x_3]]>`
` <p>Let's analyze the equations in the system represented by this augmented matrix.  The third equation will read \$0=1\$.  This is patently false, all the time.  No choice of values for our variables will ever make it true.  We're done.  Since we cannot even make the last equation true, we have no hope of making all of the equations simultaneously true.  So this system has no solutions, and its solution set is the empty set, \$\emptyset=\set{\ }\$ (<acroref type="definition" acro="ES" />).</p>`
` `
` <p>Notice that we could have reached this conclusion sooner.  After performing the row operation`
`-\$\rowopadd{-7}{2}{3}\$, we can see that the third equation reads \$0=-5\$, a false statement.  Since the system represented by this matrix has no solutions, none of the systems represented has any solutions.  However, for this example, we have chosen to bring the matrix fully  to reduced row-echelon form for the practice.</p>`
`+\$\rowopadd{-7}{2}{3}\$, we can see that the third equation reads \$0=-5\$, a false statement.  Since the system represented by this matrix has no solutions, none of the systems represented has any solutions.  However, for this example, we have chosen to bring the matrix all the way  to reduced row-echelon form as practice.</p>`
` `
` </example>`
` `
` <p>These three examples`
` (<acroref type="example" acro="SAB" />, <acroref type="example" acro="SAA" />, <acroref type="example" acro="SAE" />)`
`-illustrate the full range of possibilities for a system of linear equations <mdash /> no solutions, one solution, or infinitely many solutions.  In the next section we'll examine these three scenarios more closely.</p>`
`+illustrate the full range of possibilities for a system of linear equations <mdash /> no solutions, one solution, or infinitely many solutions.  In the next section we will examine these three scenarios more closely.</p>`
` `
`-<p>We (and everybody else) will often speak of <q>row-reducing</q> a matrix.  This is an informal way of saying we want to begin with a matrix \$A\$ and then analyze <em>the</em> matrix \$B\$ that is row-equivalent to \$A\$ and in reduced row-echelon form.  So the term <define>row-reduce</define> is used as a verb, but describes something a bit more complicated, since we do not really change \$A\$.   <acroref type="theorem" acro="REMEF" /> tells us that this process will always be successful and <acroref type="theorem" acro="RREFU" /> tells us that \$B\$ will be unambiguous.  Typically, an investigation of \$A\$ will proceed by analyzing \$B\$ and applying theorems whose hypotheses include the row-equivalence of \$A\$ and \$B\$, and usually the hypothesis that \$B\$ is in reduced row-echelon form.</p>`
`+<p>We (and everybody else) will often speak of <q>row-reducing</q> a matrix.  This is an informal way of saying we begin with a matrix \$A\$ and then analyze <em>the</em> matrix \$B\$ that is row-equivalent to \$A\$ and in reduced row-echelon form.  So the term <define>row-reduce</define> is used as a verb, but describes something a bit more complicated, since we do not really change \$A\$.   <acroref type="theorem" acro="REMEF" /> tells us that this process will always be successful and <acroref type="theorem" acro="RREFU" /> tells us that \$B\$ will be unambiguous.  Typically, an investigation of \$A\$ will proceed by analyzing \$B\$ and applying theorems whose hypotheses include the row-equivalence of \$A\$ and \$B\$, and usually the hypothesis that \$B\$ is in reduced row-echelon form.</p>`
` `
` <sageadvice acro="RREF" index="reduced row-echelon form">`
` <title>Reduced Row-Echelon Form</title>`