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\centerline{Mathematics 244 -- Linear Algebra}
\centerline{Discussion 3 -- Linear Equations, revisited}
\centerline{March 17, 2004}
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\noindent
{\it Background}
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Today, we want to think through some of the connections between 
our work with systems of linear equations from Chapter 1 and 
the language of linear mappings, kernels, images, etc.  from 
Chapter 2.  To begin, recall from the end of class on Monday, that
we have seen:
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\item{$\bullet$} If $T : {\bf R}^n \to {\bf R}^n$ is the linear
mapping whose matrix with respect to the standard bases is 
$$[T] = A = \pmatrix{a_{11} & a_{12} & \cdots & a_{1n}\cr
                 a_{21} & a_{22} & \cdots & a_{2n}\cr
                 \vdots &\vdots & \ddots & \vdots \cr
                 a_{m1} & a_{m2} & \cdots & a_{mn}\cr},$$
${\bf x} = (x_1, x_2, \ldots, x_n)^t$ is a vector in ${\bf R}^n$, 
and ${\bf b} = (b_1, b_2, \ldots, b_m)^t$ is a vector in ${\bf R}^m$, then
the equation $T({\bf x}) = {\bf b}$ is equivalent to the system of
linear equations:
$$\matrix{
a_{11} x_1 & + & a_{12} x_2 & + \cdots + & a_{1n} x_n & = & b_1\cr
a_{21} x_1 & + & a_{22} x_2 & + \cdots + & a_{2n} x_n & = & b_2\cr
    \vdots &     &\vdots&      & \vdots & \vdots & \vdots\cr
a_{m1} x_1 & + & a_{m2} x_2 & + \cdots + & a_{mn} x_n & = & b_m\cr}.
\leqno(1)$$
\item{$\bullet$} The system of equations (1) can be written in the 
shorthand form $A{\bf x} = {\bf b}$.  
\item{$\bullet$} Hence, ${\bf x} \in {\rm Ker}(T)$ if and only if 
$A {\bf x} = {\bf 0}$ -- that is, if and only if 
${\bf x}$ is a solution of the system (1) with 
$b_1 = \cdots = b_m = 0$.
\item{$\bullet$} ${\bf b}$ is in ${\rm Im}(T)$ if and only if
the system (1) for that ${\bf b}$ {\it has at least one solution}.
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\noindent
{\it Discussion Questions}
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\noindent
A)  Consider the echelon form system of equations $A{\bf x} = {\bf b}$
defined by 
$$A = \pmatrix{1 & 3 & 0 & -1 & 2\cr
               0 & 0 & 1 &  3 & 1 \cr}$$
and ${\bf b} = (3,7)^t$ (i.e. the column vector with these components).
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\item{1)}  Find parametrization of the set of solutions of this system.
\item{2)}  Show that your set of solutions is the same as the set
of vectors in $(3,0,7,0,0) + {\rm Ker}(T)$ (the sum of a fixed vector
and a subspace -- this notation is the same as Exercise 5 from \S 1.2.)
\item{3)}  Verify that the Dimension Theorem holds here.
\item{4)}  Explain why this says that {\it the general solution}
of $A{\bf x} = {\bf b}$ is any {\it particular solution}, plus
the general solution of the corresponding homogeneous system
$A {\bf x} = {\bf 0}$.
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\noindent
B)  The pattern from part A is true in general.  Suppose that 
$T : V \to W$ is any linear mapping, and assume that ${\bf b} \in {\rm Im}(T)$
so there is some ${\bf x}_0$ such that $T({\bf x}_0) = {\bf b}$.
Then show that the set $S$ of all solutions ${\bf x}$ of the equation 
$T({\bf x}) = {\bf b}$ is the set
$$S = {\bf x}_0 + {\rm Ker}(T)$$
(Hint: Show $S \subseteq {\bf x}_0 + {\rm Ker}(T)$ and 
${\bf x}_0 + {\rm Ker}(T) \subseteq S$.)
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\noindent
C)  The result from parts A and B also holds in situations like the 
following.  Suppose we want to find all solution functions $y = y(x)$ of the
differential equation 
$${d^2y \over dx^2} - e^x = 0\leqno(2)$$
Let $T : C^2({\bf R}) \to C({\bf R})$ be the mapping
$T(f) = {d^2f\over dx^2}$.  What is the kernel of $T$?  Find the    
general solution of (2).  Does the Dimension Theorem apply
in this example?  Why or why not?
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\noindent
D)  The matrix-vector form $A{\bf x} = {\bf b}$ of the system (1)
can also be used to formulate a ``shortcut'' method for the process
of reducing a system of equations to echelon form and identifying 
the set of solutions.  The {\it augmented matrix} of (1) is the 
$m \times (n+1)$ matrix
$$(A\ |\ {\bf b}) =  \pmatrix{a_{11} & a_{12} & \cdots & a_{1n} &|\ b_1\cr
                 a_{21} & a_{22} & \cdots & a_{2n} &|\ b_2\cr
                 \vdots &\vdots & \ddots & \vdots & \vdots\cr
                 a_{m1} & a_{m2} & \cdots & a_{mn} &|\ b_m\cr}$$
(just ``tack'' ${\bf b}$ ``onto $A$'' as an additional column).
All of the arithmetic of reducing to the echelon form system
can be done directly on this matrix (i.e. without explicitly writing the 
variables $x_j$ for $j = 1,\ldots, n$ at each step).  To see how this 
works, find the augmented matrix of the following system
$$\matrix{&3x_1 & - & 2 x_2 & + &x_3 & = & 1\cr
          &x_1 & - & x_2 & + & 4x_3 & = & 0\cr}$$
carry out the operations to reduce to echelon forms directly 
on the rows of the matrix, and read off a parametrization of 
the set of solutions.  {\it Also,} explain how your answer relates
to the Dimension Theorem.
    
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{\it Assignment}
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Group writeups due on Monday, March 22.
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