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Linear Equations and Regular Matrices
In general, a system like (1) may have no solutions (in
which case we
say that the system is inconsistent or overdetermined), infinitely many solu-
tions (in which case we say that the system is underdetermined), or exactly
one solution. If , then we say that the system (1) is
Exercise 1 Show that is always a solution of a homoge -
neous system and conclude that a homogeneous systems has infinitely many
solutions if and only if it has a non- zero solution .
The coefficient matrix of (1) is the matrix
and the extended or augmented matrix of (1) is the matrix
Exercise 2 Find the coefficient matrix and the
augmented matrix for the
following system of linear equations :
and find the homogeneous system with the same coefficient
matrix, as well
as its augmented matrix.
The columns of B can be written as n + 1 column vectors
A linear combination of vectors
is a vector
that can be
where are scalars,
that is, numbers.
In the notation of (5), the system (1) can be written as
To see this, note that
and the last line of (8) is equal to
It can be shown that
is a linear combination of . That is, there
exist numbers such that . Write down
a system of linear equations whose solution will give you these numbers
. What is the coefficient matrix A for this system?
A set of vectors is linearly dependent if one of these vectors can
be expressed as a linear combination of the other vectors. For example, if
, then the vectors are linearly dependent. An
equivalent definition that is more commonly used in the literature says that
are linearly dependent if there exist scalars (numbers) ,
not all of them zero, so that
This latter definition gives an interesting connection
with solutions of
homogeneous linear systems. Note that (1) is homogeneous if and only if
. Thus a homogeneous system (1) has a non-zero solution if and only
if the columns of the coefficient matrix A are linearly dependent.
Exercise 4 Consider the vectors
Show that these vectors are linearly dependent in two
different ways :
(a) Pick one of these vectors and express it as a linear combination of the
remaining three vectors.
(b) Find numbers , not all of them zero, such that
Vectors that are not linearly dependent are called linearly independent.
By rephrasing the second definition of linear dependence we can see that
vectors are linearly independent if and only if the equality
implies that . The
connection between linear de-
pendence and solutions of homogeneous systems of linear equations that
we mentioned above can now be rephrased as follows: A homogeneous sys-
tem (1) has no non-zero solution if and only if the columns of the
coefficient matrix A are linearly independent. The double negative in the
last sentence becomes easier to understand if we think about it this way: In
Exercise 1 you showed that the zero vector is always a solution of a homo-
geneous system (1). It is the unique solution of this system if and only if
the column vectors of its coefficient matrix A are linearly independent.
Thus if we want to find out whether a (homogeneous) system of linear
equations (1) has a unique solution, we will need to find out whether the
column vectors of its coefficient matrix are linearly independent. It is usually
quite difficult to do this directly from the definition (as you may have found
out while doing Exercise 4). Fortunately, if A is a square matrix , that
is, if A has dimension n × n for some n, there is a shortcut: For every
square matrix A, one can compute a number det (A) called the determinant
of A, and this number tells us whether the column vectors of A are linearly
independent. More precisely , the following holds: