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Definition
An equation such as x+3y=9 is called a linear equation
(in two variables or unknowns).
The graph of this equation is a straight line in the xy-plane.
A pair of values of x and y that satisfy the equation is called a solution.
1.1 Matrices and Systems of Linear Equations

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Definition
A linear equation in n variables x1, x2, x3, …, xn has the form a1 x1 + a2 x2 + a3 x3 + … + an xn = b
where the coefficients a1, a2, a3, …, an and b are real numbers.

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Solutions for system of linear equations
Figure 1.1
Unique solution
x + 3y = 9
–2x + y = –4
Lines intersect at (3, 2)
Unique solution: x = 3, y = 2.

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A linear equation in three variables corresponds to
a plane in three-dimensional space.
※ Systems of three linear equations in three variables:

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A solution to a system of a three linear equations will be points that lie on all three planes.
The following is an example of a system of three linear equations:
How to solve a system of linear equations? For this we introduce a method called Gauss-Jordan elimination. (Section1.2)

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Definition
A matrix is a rectangular array of numbers.
The numbers in the array are called the elements of the matrix.
Matrices

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Submatrix
Row and Column

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matrix of coefficients and augmented matrix
Relations between system of linear equations and matrices

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Elementary Transformation
Interchange two equations.
Multiply both sides of an equation by a nonzero constant.
Add a multiple of one equation to another equation.
Elementary Row Operation
Interchange two rows of a matrix.
Multiply the elements of a row by a nonzero constant.
Add a multiple of the elements of one row to the corresponding elements of another row.
Elementary Row Operations of Matrices

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Example 1
Solving the following system of linear equation.

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Example 2
Solving the following system of linear equation.
Solution

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Example 3
Solve the system
Solution

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Summary
Use row operations to [A: B] :
Def. [In : X] is called the reduced echelon form of [A : B].
Note. 1. If A is the matrix of coefficients of a system of n equations in n variables that has a unique solution,
then A is row equivalent to In (A ≈ In).
2. If A ≈ In, then the system has unique solution.

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Example 4 Many Systems
Solving the following three systems of linear equation, all of which have the same matrix of coefficients.
Solution
The solutions to
the three systems are

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Homework
Exercises will be given by the teachers of the practical classes.

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1.2 Gauss-Jordan Elimination
Definition
A matrix is in reduced echelon form if
Any rows consisting entirely of zeros are grouped at the bottom of the matrix.
The first nonzero element of each other row is 1. This element is called a leading 1.
The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row.
All other elements in a column that contains a leading 1 are zero.

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Examples for reduced echelon form
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elementary row operations，reduced echelon form
The reduced echelon form of a matrix is unique.

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Gauss-Jordan Elimination
System of linear equations
⇒ augmented matrix
⇒ reduced echelon form
⇒ solution

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Example 1
Use the method of Gauss-Jordan elimination to find reduced echelon form of the following matrix.
Solution
The matrix is the reduced echelon form of the given matrix.

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Example 2
Solve, if possible, the system of equations
Solution
The general solution to the system is

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Example 3
Solve the system of equations
Solution
⇒ many sol.

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Example 4
Solve the system of equations
Solution

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Example 5
This example illustrates a system that has no solution. Let us try to solve the system
Solution
The system has no solution.

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Homogeneous System of linear Equations
Definition
A system of linear equations is said to be homogeneous if all the constant terms are zeros.

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Homogeneous System of linear Equations
Note. Non trivial solution

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Homework
Exercise will be given by the teachers of the practical classes.

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1.3 Gaussian Elimination
Definition
A matrix is in echelon form if
Any rows consisting entirely of zeros are grouped at the bottom of the matrix.
The first nonzero element of each row is 1. This element is called a leading 1.
The leading 1 of each row after the first is positioned to the right of the leading 1 of the previous row.
(This implies that all the elements below a leading 1 are zero.)

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Example 6
Solving the following system of linear equations using the method of Gaussian elimination.
Solution
We have arrived at the echelon form.

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The corresponding system of equation is

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Example 7
Solving the following system of linear equations using the method of Gaussian elimination, performing back substitution using matrices.
Solution
This marks the end of the forward elimination of variables from equations. We now commence the back substitution using matrices.