Guide on System of Linear Equations in Linear Algebra
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Linear equation
An equation is said to be a linear equation with
Where:
, , , and are some constants.at least one of
, , , is non-zero.
The variables
Linear equations
The following is an example of a linear equation with two variables:
Note the following:
when working with two variables, we often denote the variables as
and .when working with three variables, we often denote the variables as
, and .
The following is an example of a linear equation with three variables:
Non-linear equations
The following are some examples of non-linear equations:
These are non-linear because the variables
Graphical representation of a linear equation
Recall that an equation of a line in two-dimensional space is given by:
Where

Similarly, a linear equation with three variables

As we can see, the linear equation with
The graphs of the linear equations motivate why linear equations are described as "linear" - their graphs are straight and non-curvy. In contrast, a non-linear equation such as

Homogeneous linear equation
A homogeneous linear equation with
A homogeneous linear system is a collection of homogeneous linear equations - for instance:
We will later cover some applications of homogenous linear equations.
System of linear equations
A finite set of linear equations with
A solution of a system of linear equations is the values that the variables
Computing a solution of a system of linear equations
Compute the solution of the following simultaneous equations:
Solution. Subtracting the bottom equation from the top equation yields:
Substituting
Therefore, the solution of the simultaneous equations is the pair
Consistent and inconsistent system of linear equations
A consistent system of linear equations has either one solution or infinitely many solutions. An inconsistent system of linear equations has no solution.
Inconsistent system with no solution
Consider the following system of linear equations:
Show that this system is inconsistent.
Solution. Subtracting the second equation from the first equation yields:
We get a contradiction here, which means that there exists no
Let's graph the two equations:

As we can see, the lines are parallel to each other and thus never intersect.
Consistent system with a single solution
Consider the following system of linear equations:
Show that this system is consistent.
Solution. Subtracting the second equation from the first equation:
Substituting the value of
Since a solution exists, this system of linear equations is consistent.
Let's graph the two equations:

We can see that the two lines intersect at a single point
Consistent system with infinitely many solutions
Consider the following system of linear equations:
Show that this system is consistent.
Solution. Multiplying the top equation by
This tells us that the top and bottom equations are equivalent! We have only
Here, setting any value for
Let's plot the two equations:

We only see one line here because the two equations overlap with one another. Every point on this line is a solution.
Matrix form of system of linear equations
All systems of linear equations can be expressed in the following matrix form:
Where:
is the matrix holding the coefficients of the variables. is a vector holding the variables , , , . is a vector holding the constant terms , , , .
Expressing a system of linear equations in matrix form allows us to use a technique called Gaussian Elimination to find the solutions. This will be the topic of the next section.
Sketch proof. Consider the following system of linear equations:
This can be written in the following matrix form:
Performing the matrix-vector multiplication on the left:
Equating the components gives us the original system of linear equations.
Here's another example - consider the following simultaneous equation:
The corresponding matrix form is:
We often denote:
the matrix containing the coefficients as
.the vector containing the variables as
.the vector on the right as
.
Therefore, the short-hand notation of the matrix form of a system of linear equations is:
This completes the sketch proof.
Expressing a system of linear equations in matrix form
Express the following system of linear equations in matrix form:
Solution. The system of linear equation in matrix form is:
The following theorem is useful for future proofs - feel free to skip it for now and come back later when referenced.
Expressing a system of vector equations as a matrix product
Proof. Suppose we have the following vector equations:
Define an
The transpose
By theoremlink, we have that:
Here, the column vectors of
By theoremlink, every column in
This completes the proof.
Practice problems
Consider the following equations:
Identify the linear equations.
All equations are linear.
None of the equations are linear.
Equation 1 is linear.
Equations 1 and 3 are linear.
Equation 2 is not linear because the variable
Solve the following system of linear equations:
Separate your values for
Multiply the bottom equation to get:
Subtracting the bottom equation from the top equation gives:
Substituting this value of
Consider the following system of linear equations:
Which of the following is true?
The system is consistent with an unique solution.
The system is consistent with infinitely many solutions.
The system is inconsistent with no solution.
The solution to this system of linear equations is
Consider the following system of linear equations:
Which of the following is true?
The system is consistent with an unique solution.
The system is consistent with infinitely many solutions.
The system is inconsistent with no solution.
Multiplying the second equation by
Consider the following system of linear equations:
Express this in matrix form.
The matrix form of the system of linear equations is:
Consider the following:
Express this as a system of linear equations in non-matrix form.
The system of linear equations in non-matrix form is: