Even at school, each of us studied equations and, probably, systems of equations. But not many people know that there are several ways to solve them. Today we will analyze in detail all the methods for solving a system of linear algebraic equations, which consist of more than two equalities.
History
Today it is known that the art of solving equations and their systems originated in Ancient Babylon and Egypt. However, equalities in their usual form for us appeared after the appearance of the equal sign "=", which was introduced in 1556 by the English mathematician Record. By the way, this sign was chosen for a reason: it means two parallel equal segments. Indeed, there is no better example of equality.
The founder of modern lettering of the unknowns and signs of degrees is the French mathematician Francois Viet. However, its designations were significantly different from today. For example, he designated a square of an unknown number with the letter Q (lat. "Quadratus"), and a cube with the letter C (lat. "Cubus"). These notations now seem uncomfortable, but then it was the most understandable way to write systems of linear algebraic equations.
However, the drawback in the then-solution methods was that mathematicians considered only positive roots. Perhaps this is due to the fact that negative values had no practical application. One way or another, it was the Italian mathematicians Niccolo Tartaglia, Gerolamo Cardano and Rafael Bombelli who began to consider negative roots first in the 16th century. And the modern look, the main method of solving quadratic equations (through discriminant) was created only in the 17th century thanks to the work of Descartes and Newton.
In the mid-18th century, Swiss mathematician Gabriel Kramer found a new way to make solving systems of linear equations easier. This method was subsequently named after him and to this day we use it. But we will talk about the Cramer method a bit later, but for now we will discuss linear equations and methods for solving them separately from the system.
Linear equations
Linear equations are the simplest equalities with a variable (s). They are classified as algebraic. The linear equations are written in general form as follows: a 1 * x 1 + a 2 * x 2 + ... a n * x n = b. We will need to represent them in this form when compiling systems and matrices below.
Systems of linear algebraic equations
The definition of this term is this: it is a set of equations that have common unknown values and a general solution. As a rule, in school everything was solved by systems with two or even three equations. But there are systems with four or more components. Let's figure out first how to write them down so that it is convenient to solve in the future. Firstly, systems of linear algebraic equations will look better if all the variables are written as x with the corresponding index: 1,2,3 and so on. Secondly, all equations should be reduced to the canonical form: a 1 * x 1 + a 2 * x 2 + ... a n * x n = b.
After all these steps, we can begin to tell how to find a solution to systems of linear equations. Matrices are very useful for this.
Matrices
A matrix is a table that consists of rows and columns, and its elements are at their intersection. These can be either specific values or variables. Most often, to indicate elements, lower indices are placed below them (for example, a 11 or a 23 ). The first index is the row number, and the second is the column. You can perform various operations on matrices, as well as on any other mathematical element. Thus, you can:
1) Subtract and add the same sized tables.
2) Multiply the matrix by some number or vector.
3) Transpose: turn the rows of the matrix into columns, and the columns into rows.
4) Multiply matrices if the number of rows in one of them is equal to the number of columns in the other.
We will discuss all these techniques in more detail, as they will be useful to us in the future. Subtraction and addition of matrices is very simple. Since we take matrices of the same size, each element of one table corresponds to each element of another. Thus, we add (subtract) these two elements (it is important that they stand at the same places in their matrices). When multiplying a matrix by a number or a vector, you just need to multiply each element of the matrix by that number (or vector). Transposition is a very interesting process. It is very interesting sometimes to see it in real life, for example, when changing the orientation of a tablet or phone. Icons on the desktop are a matrix, and when you change the position, it is transposed and becomes wider, but decreases in height.
We will also consider such a process as matrix multiplication. Although it is not useful to us, it will still be useful to know him. Two matrices can only be multiplied provided that the number of columns in one table is equal to the number of rows in the other. Now take the row elements of one matrix and the elements of the corresponding column of another. Multiply them by each other and then add them (that is, for example, the product of the elements a 11 and a 12 by b 12 and b 22 will be equal to: a 11 * b 12 + a 12 * b 22 ). Thus, one element of the table is obtained, and in the same way it is filled further.
Now we can begin to consider how the system of linear equations is solved.
Gauss method
This topic is beginning to take place at school. We well know the concept of "system of two linear equations" and are able to solve them. But what if the number of equations is more than two? The Gauss method will help us with this .
Of course, this method is convenient to use if you make a matrix out of the system. But you can not transform it and decide in its purest form.
So, how is the system of linear Gauss equations solved by this method? By the way, even though this method was named after him, they discovered it in ancient times. Gauss suggests the following: carry out operations with equations in order to eventually bring the entire population to a stepwise form. That is, it is necessary that from top to bottom (if correctly placed) from the first equation to the last one decreases by one unknown. In other words, we need to make sure that we get, say, three equations: in the first - three unknowns, in the second - two, in the third - one. Then from the last equation we find the first unknown, substitute its value in the second or first equation, and then find the remaining two variables.
Cramer Method
To master this method, it is vital to have the skills of addition, subtraction of matrices, and also you need to be able to find determinants. Therefore, if you do it badly or do not know how to do it at all, you will have to learn and practice.
What is the essence of this method, and how to make a system of linear Cramer equations? Everything is very simple. We must build a matrix of numerical (almost always) coefficients of a system of linear algebraic equations. To do this, simply take the numbers in front of the unknowns and arrange them in a table in the order they are written in the system. If the number is preceded by a "-" sign, then we write a negative coefficient. So, we compiled the first matrix of coefficients for unknowns, not including numbers after equal signs (naturally, the equation should be reduced to canonical form, when only the number is on the right and all unknowns with coefficients on the left). Then you need to make a few more matrices - one for each variable. To do this, we replace each column with coefficients in the first matrix in turn with a column of numbers after the equal sign. Thus, we obtain several matrices and then find their determinants.
After we found the qualifiers, it’s up to the small. We have an initial matrix, and there are several resulting matrices that correspond to different variables. To get the system solutions, we divide the determinant of the resulting table by the determinant of the initial table. The resulting number is the value of one of the variables. Similarly, we find all the unknowns.
Other methods
There are several more methods in order to obtain a solution of systems of linear equations. For example, the so-called Gauss-Jordan method, which is used to find solutions of a system of quadratic equations and is also associated with the use of matrices. There is also a Jacobi method for solving a system of linear algebraic equations. It is easiest to adapt to a computer and is used in computer technology.
Difficult cases
Complexity usually arises if the number of equations is less than the number of variables. Then we can certainly say that either the system is incompatible (that is, it has no roots), or the number of its solutions tends to infinity. If we have the second case, then we need to write down the general solution of the system of linear equations. It will contain at least one variable.
Conclusion
So we have come to an end. To summarize: we examined what a system and a matrix are, learned how to find a general solution to a system of linear equations. In addition, we considered other options. We found out how the system of linear equations is solved: the Gauss method and the Cramer method. We talked about difficult cases and other ways of finding solutions.
In fact, this topic is much more extensive, and if you want to better understand it, we recommend reading more specialized literature.