Homogeneous and Nonhomogeneous Systems
A homogeneous system of linear equations is one
in which all of the constant terms are zero. A homogeneous system always has
at least one solution, namely the zero vector. When a row operation is
applied to a homogeneous system, the new system is still homogeneous. It is
important to note that when we represent a homogeneous system as a matrix,
we often leave off the final column of constant terms, since applying
row operations would not modify that column. So, we use a regular matrix
instead of an augmented matrix. Of course, when looking for a solution,
it's important to take the constant zero terms into account.
A nonhomogeneous system has an associated homogeneous system,
which you get by replacing the constant term in each equation with zero.
Section 1.I.3 in the textbook is about understanding the structure of
solution sets of homogeneous and non-homogeneous systems. The main theorems
that are proved in this section are:
Theorem: The solution set of
a homogeneous linear system with variables is of the form
,
where is the number of free variables in an echelon form of
the system and
are [constant] vectors in
Theorem: Consider a system of linear equations
in variables, and suppose that is a solution of the system. Then
the solution set of the system is of the form
,
where
is the solution set of the associated homogeneous system. (So, is still the
number of free variables in an echelon form of the system.)
The vector in the second theorem is said to be a
particular solution of the system. (Remember that for a nonhomogeneous
system, it is possible that no particular solution exists, and the solution set is empty.)
A homogeneous system always has as a particular solution, and the second
theorem applies to homogeneous systems by taking .
Note that for a given system, the vectors and are not unique.
There can be many different sequences of row operatation that could be used
to put the system into echelon form. The and that you get
can depend on the specific sequence of row operations that you use. However, you
can only get different ways of writing the same solution set. (A possibly
surprising fact, which has not yet been proved, is that no matter what sequence
of row operations you use to put the system into echelon form, you always get
the same number of free variables. This means that the number in the
system is uniquely determined by the system.)
These theorems do not really change the way that you go about solving a linear system,
but it does help us understand the structure of the solution set of the system,
and in particular the geometry of the solution set. The solution set of a
homogeneous system with variables is a linear space in that
contains the origin. Adding the
vector to all the points in that linear space gives a "parallel" linear
space that contains . See the
second picture
in the previous sectio].
The other significant topic in Section 1.I.3 of the textbook is
singular versus nonsingular matrices. This is an issue only for square
matrices. (A square matrix is one in which the
number of rows is equal to the number of columns.) A square matrix is the
associated matrix of some homogeneous system. Since the matrix is square,
the homogeneous system has the same number of equations as there are
variables. The homogeneous system will either have as its only solution, or
it will have an infinite number of solutions. The matrix is said to be
nonsingular if the system has a unique solution.
It is said to be singular if the system has an infinite
number of solutions. (The terms "singular" and "nonsingular" only
apply to square matrices.)
Note that, by the above theorems, a square matrix
is singular if and only it has at least one free variable when it is put
into echelon form, which in turn is true if and only if an echelon
form of the matrix has at least one row containing only zeros.
I would like to add some geometric perspective to all this, using
ideas from the previous section.
You should try to develop your higher-dimensional geometric intuition!
Suppose we have a homogeneous system of equations in variables.
The solution set for each individual equation in the system is a
linear space in of dimension of (unless the equation is the
trivial equation, .) And since is a solution, that linear
space passes through the origin. The solution set for the entire system is
the intersection of the solution sets for all of the individual equations;
that is, it is the intersection of linear spaces of dimension .
For example, if ,
we are looking at the intersection of two lines through the origin; the possibilities
are that the lines intersect only at the origin [nonsingular matrix] or that
the lines are actually identical [singular matrix]. Of course, if you pick
two lines at random, it will be very unlikely that they are identical.
That means that if you pick a matrix at random, it is very unlikely
that it will be singular.
For , we are intersecting three planes that contain the origin. The
intersection of two planes through the origin is a line, unless the planes happen
to be identical. When you add the third plane to the intersection, you are
most likely intersecting that plane with a line and the result will be a single
point (namely the origin), except in the unlikely case that the line happens
to lie entirely in the plane.
For dimension , the situation is similar. The solution set to the
first equation is a linear space in of dimension .
Taking the intersection of that space with the solution set of the
second equation is likely to give a linear space of dimension .
As the solution set for each equation is added to the intersection, the
dimension of the intersection is likely to go down by one. When you get
to the intersection of the solution sets for all equations, you are
likely to have a space of dimension zero—a single point, namely the
origin. Again, if you pick an matrix at random, it is very
unlikely to be singular. (The word "singular" means "notably unusual.")
When you look a nonhomogeneous linear system of equations in variables,
you are intersecting solution sets that do not necessarily contain the origin.
The mostly likely possibility for the intersection is still a single point,
and an infinite intersection is still possible. But you also have the possiblility
of an empty intersection—no solution—as would happen for example if
you intersect two parallel lines.
We can also think about what happens when we apply row reduction to
put an matrix into echelon form. Consider the matrix as the
matrix for a homogeneous linear system, written without the constant zero
terms from the right sides of the equations.
Remember that the matrix is square, with the same number of rows as of columns.
The echelon form matrix that results from row reduction will have the form
where all the entries below the diagonal are zero (and some of the could also be zero). But there might be
all-zero rows at the bottom. That is, might be zero.
If , then there are no free variables, and the homogeneous
system has as its only solution. If , there
is at least one non-zero row, and at most of the rows are non-zero.
So, there are at most leading variables, which implies that there is at least one free
variable; the system has an infinite number of solutions.
Now, suppose that we have a nonhomogeneous system with the same
matrix of coefficients. The augmented matrix for the nonhomogeneous
system has the form
If we apply Gauss's method using the same row operations that we
used for the homogeneous system, we get a matrix of the form
Now, in the case , there are no free variables and we can solve the system to get
a unique solution. (Thus, a linear system whose matrix of coefficients
is a square, nonsingular matrix will always have a unique solution.)
In the case , we have to consider the
constant terms. When and , we have an equation
of the form where , and there is no solution. When
and , we have an all-zero row. However,
there might still be no solution because one of the previous equations
might still be of the form , where . However, we can
say that if there are no such rows, then we have a solvable system with
at least one free variable, and the number of solutions is infinite.
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