1.1: System of Linear equations
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Introduction to System of Linear equations
Under Construction!
Definition: Linear Equations
a1x1+a2x2+⋯+anxn=b is called a linear equation in n variables x1,x2,⋯,xn, where ai∈R is the coefficient of xi, for i=1,⋯,n and b∈R is the constant term.
Example 1.1.1
- x=2 lis a inear equation in one variable x.
- x+y=3 is a linear equation in two variables x and y.
- x2+y=3 is not a linear equation.
Example 1.1.2
Find two numbers whose sum is 12 and whose positive difference is 3.
Solution
Let x and y be numbers such that x+y=12 and x−y=3. This is a system of two linear equations in two variables.
Then we can solve these equations by eliminating one variable.
By adding the equations, we get 2x=15. Therefore, x=152=7.5.
Now, y=12−x=12−7.5=4.5.
Thus the numbers are 7.5 and 4.5. Thus the system has a unique solution.
We can also find the solution using geometry. Remember linear equations in two variables represent lines in two dimensions. From the graph in Figure 1.1.1, we can see that the intersection point is (7.5,4.5).
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Definitions:
1. a1x1+a2x2+⋯+anxn=0 is called a homogeneous linear equation in n variables x1,x2,⋯,xn.
2. A system of linear equations in the variables, x1,x2⋯,xn, is a finite collection of linear equations. a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮am1x1+am2x2+⋯+amnxn=bm where aij and bj are real numbers. The above is a system of m equations in the n variables, x1,x2⋯,xn. Written more simply in terms of summation notation, the above can be written in the form n∑j=1aijxj=bi, i=1,2,3,⋯,m
3. A system of linear equations is called a homogeneous system if the constant term of each equation in the system is equal to 0. A homogeneous system has the form ca11x1+a12x2+⋯+a1nxn=0a21x1+a22x2+⋯+a2nxn=0⋮am1x1+am2x2+⋯+amnxn=0 where aij are scalars and xi are variable
4. A list (s1,s2,⋯,sn) is a solution of the system of linear equations if it satisfies all the linear equations in the system.
Example 1.1.3
x+y−3z=32x−y=44x+2y+3z=7
Solutions to System of Linear equations
Possible solutions:
Consider the following systems of two equations in two variables.
x−y=0x+y=0}
x−y=0x−y=1}
x−y=02x−2y=0}
What are the solutions to the above systems.
Definition:
A system of linear equations is called consistent if there exists at least one solution. It is called inconsistent if there is no solution.
Augmented matrix
Definition: Elementary Row operations
The elementary row operations consist of the following
1. Interchange two rows.
2. Multiply a row by a nonzero number.
3. Replace a row with any multiple of another row added to it.