Systems of linear equations

A linear equation is an equation of the form

               a1x1+a2x2+...+anxn=b             (1)

where x₁,...,x_n are unknowns, a₁,...,a_n, b are coefficients.

Example:

This equation has three unknowns and four coefficients (3, -4, 5, 6).

A solution of a linear equation (1) is a sequence of numbers x₁,...,x_n which make (1) a true equality.

Example:

is a solution of equation (2).

A linear equation can have infinitely many solutions, exactly one solution or no solutions at all.

Equation (2) has infinitely many solutions. To find them all we can set arbitrary values of x and y and then solve (2) for z.

We get:

				               x=s
				               y=t
				               z=(6-3s+4t)/5

These formulas give all solutions of our equation meaning that for every choice of values of t and s we get a solution and every solution is obtained this way. Thus this is a (the) general solution of our equation.

There may be many formulas giving all solutions of a given equation. For example Maple gives another formula:
> with(linalg);

This command starts the linear algebra package.

> A:=matrix(1,3,[3,-4,5]):
b:=vector([6]):
linsolve(A, b);

This command asks Maple to solve the system of equations.

The solution has two parameters t₁ and t₂;

In order to get this solution "by hand" one can give y and z arbitrary values (t₁ and t₂) and solve for x.

A system of linear equations is any sequence of linear equations. A solution of a system of linear equations is any common solution of these equations. A system is called consistent if it has a solution. A general solution of a system of linear equations is a formula which gives all solutions for different values of parameters.

Examples. 1. Consider the system:

					          x +  y = 7
					         2x + 4y = 18

This system has just one solution: x=5, y=2. This is a general solution of the system.

2. Consider the system:

					       x +  y + z = 7
					      2x + 4y + z = 18.

This system has infinitely many solutions given by this formula:

					        x = 5-3s/2
					        y = 2+s/2
					        z = s

This is a general solution of our system.

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In order to find a general solution of a system of equations, one needs to simplify it as much as possible. The simplest system of linear equations is

					        x = a
					        y = b
					         ...

where every equation has only one unknown and all these unknowns are different. It is not possible to reduce every system of linear equations to this form, but we can get very close. There are three operations that one can apply to any system of linear equations:

1. Replace an equation by the sum of this equation and another equation multiplied by a number.
2. Swap two equations.
3. Multiply an equation by a non-zero number.

The system obtained after each of these operations is equivalent to the original system, meaning that they have the same solutions.

For example consider the system

					       x +  y = 7
					      2x + 4y = 18

We can first replace the second equation by the second equation plus the first equation multiplied by -2. We get

					       x + y = 7
					          2y = 4

Now we can use the third operation and multiply the second equation by 1/2:

					       x + y = 7
					           y = 2

Finally we can replace the first equation by the sum of the first equation and the second equation multiplied by -1:

					          x = 5
					          y = 2

Since this system is equivalent to the original system, we get that x=5, y=2 is the general solution of the original system.

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