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\centerline{Mathematics 244 -- Linear Algebra}
\centerline{Discussion 2 -- More on Systems of Linear Equations}
\centerline{February 11, 2004}
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{\it Background}
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We have seen a systematic approach to solving systems of 
linear equations by applying elementary operations to put
the system into {\it echelon form}.  We can then parametrize
the set of solutions (if there are any) by solving for 
the {\it basic variables} of the system (the ones appearing
in leading terms in the echelon form system) in terms of the 
free variables.  In the text, we often substitute values
or other parameters for the free variables to get the 
parametrization.  Today we want to do some more examples
and think about general patterns here.
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{\it Discussion Questions}
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A)  Put the following system into echelon form:
$$\matrix{&x_1 &+ &3x_2 &+ &x_3 &= &1\cr
          &x_1 &- &x_2  &+ &x_3 &= &2\cr
          &3x_1&+ &x_2  &+ &3x_3&= &4\cr}$$
What do your results mean?  Explain why it would be reasonable
to call this system {\it inconsistent}.
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B)  In this problem we will start to consider a connection between
homogeneous and inhomogeneous systems of equations:
\item{1)}  Find a linearly independent set $S$ which spans the subspace 
$W$ of ${\bf R}^5$
consisting of all solutions of the following system of equations:
$$\matrix{&2x_1 &+ &3x_2 &+ &    &+ &x_4 &- &x_5&= &0\cr
          &x_1  &  &     &+ &x_3 &  &    &+ &x_5&= &0\cr
          &x_1  &+ &x_2  &+ &x_3 &- &x_4 &  &   &= &0\cr}$$
\item{2)} Now consider a system of equations with the same left
hand sides, but the following nonzero right hand sides:
$$\matrix{&2x_1 &+ &3x_2 &+ &    &+ &x_4 &- &x_5&= &1\cr
          &x_1  &  &     &+ &x_3 &  &    &+ &x_5&= &2\cr
          &x_1  &+ &x_2  &+ &x_3 &- &x_4 &  &   &= &3\cr}$$
Find a parametrization for the set of solutions.
\item{3)}  How are your answers in parts 1 and 2 related?  
Try to relate this to the construction of ${\bf y} + W$ from
problem 5 of Section 1.2 in the book (from Problem Set 1).
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C)  Decide whether each of the following statements is true or
false.  For the true ones, give a proof.  For the false ones,
give a counterexample.
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\item{1)}  Every homogeneous system of linear equations with
more variables than equations (that is, $m < n$)
has a solution different from the zero vector.
\item{2)}  Every system of linear equations has at least 
one free variable.
\item{3)}  If the number of equations is greater than
the number of variables ($m > n$), then a system of linear
equations has no solutions. 
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