In this lecture we show how matrices and vectors can be used to represent and analyze systems of linear equations.
A system of
linear equations in
unknowns is a set of
equations![[eq1]](/images/systems-of-linear-equations-and-matrices__3.png)
where
are the
unknowns, and
(for
and
)
and
(for
)
are known constants.
The unknowns are the values that we would like to find. Solving a system of
linear equations means finding a set of values for
such that all the equations are satisfied. Such a set is called a solution of
the system.
Example
Define the system
It
is a system of 2 equations in 2 unknowns. A solution of the system
iswhich
can be verified by substituting these two values into the
system:![[eq6]](/images/systems-of-linear-equations-and-matrices__14.png)
In general, a solution is not guaranteed to exist. If it exists, it is not guaranteed to be unique. Therefore, the theory of linear equations is concerned with three main aspects:
deriving conditions for the existence of solutions of a linear system;
understanding whether a solution is unique, and how multiple solutions are related to each other;
finding techniques that allow us to find solutions of a linear system.
The above system of
linear equations in
unknowns can be represented compactly by using matrices as
follows:where:
is the
vector of unknowns
;
is the
matrix of coefficients, whose
-th
element
is the constant that multiplies
in the
-th
equation of the system;
is the
vector of constants
.
To understand how the representation works, notice that
is a
vector whose
-th
element is equal to the dot
product of the
-th
row of
and
,
that
is,![[eq10]](/images/systems-of-linear-equations-and-matrices__36.png){kind=small}
Therefore,![[eq11]](/images/systems-of-linear-equations-and-matrices__37.png)
Example
The system
can
be represented
aswhere
the
matrix of coefficients
is![[eq14]](/images/systems-of-linear-equations-and-matrices__41.png)
the
vector of unknowns
isand
the
vector of constant terms
is
By writing a system of linear equations in matrix form, we can easily provide general conditions for the existence of a solution.
Proposition
The linear system
has
a solution if and only if
belongs to the span of the columns
of
.
The product
can
be interpreted as a linear combination of the columns of
,
with coefficients taken from
.
Therefore, the problem of solving the system is tantamount to finding a vector
of coefficients
that allows us to write
as a linear combination of the columns of
.
But
can be written as a linear combination of the columns of
if and only if it belongs to their span.
We now give a general condition for the uniqueness of the solution.
Proposition
If the linear system
has
a solution, then the solution is unique if and only if the columns of
are linearly
independent
Let's first prove the if part. We have
proved above that there is a solution if and only if
belongs to the span of the columns of
.
If the columns of
are linearly independent, then they form a
basis for their span.
Furthermore, the representation of any vector of the span as a linear
combination of the basis is unique. Therefore, if the columns of
are
linearly independent, there is only one linear combination of them that gives
as a result, that is, the solution of the system is unique. Let's now prove
the only if part. We are going to prove that if the columns are not
independent, then there is more than one solution. Let
be a solution, that
isWhen
the columns of
are linearly dependent, there exists a non-zero vector
that
satisfiesAs
a consequence, there are infinite solutions because
is a solution of the system for any scalar
:![[eq21]](/images/systems-of-linear-equations-and-matrices__72.png)
The following proposition about multiple solution holds.
Proposition
If the linear system
has
a solution and the columns of
are not linearly independent, then there are infinite solutions.
See the previous proof.
Below you can find some exercises with explained solutions.
Find the matrix representation of the system
![[eq23]](/images/systems-of-linear-equations-and-matrices__75.png)
The system can be represented
aswhere
the
matrix of coefficients
is![[eq25]](/images/systems-of-linear-equations-and-matrices__78.png)
the
vector of unknowns
isand
the
vector of constant terms
is
Define![[eq28]](/images/systems-of-linear-equations-and-matrices__83.png)
Write
down the equations of the
system
The two equations of the systems
are
Please cite as:
Taboga, Marco (2021). "Systems of linear equations and matrices", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/systems-of-linear-equations-and-matrices.
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