The learning breakthrough point of linear algebra: linear equations. In other words, linear algebra can be regarded as a discipline established in the process of studying linear equations.
Characteristics of linear equations: the equations are homogeneous forms of unknowns, and the number s of equations and the number n of unknowns can be the same or different.
There are three problems worth discussing about the solution of linear equations:
(1), whether the equations have a solution, that is, the existence of a solution;
(2) How to solve the equations and how many solutions there are;
(3) When a system of equations has multiple solutions, is there any internal relationship between these different solutions, that is, the structure of the solutions?
Gauss elimination is the most basic and direct method to solve linear equations, which involves three homotopy transformations of equations:
(1), multiplying k of one equation by another equation;
(2) Exchange the positions of the two equations;
(3) Multiply an equation by a constant k. We call these three transformations elementary transformations of linear equations.
Any linear equation can be transformed into a trapezoidal equation by elementary transformation.
It can be seen from specific examples that after being transformed into trapezoidal equations, the value of each unknown can be solved in turn, thus the solution of the equations can be obtained.
It is the coefficients and their relative positions that play a decisive role in solving the equation, so we can extract all the coefficients and constant terms of the equation according to their original positions and form a table. By studying this table, we can judge the situation of the solution. We call such a table composed of several numbers in some way a matrix.
Linear equations can be expressed in matrix form, at least in writing and expression.
Coefficient matrix and augmented matrix.
The elementary transformation of linear equations in Gaussian elimination corresponds to the elementary row transformation of matrix. The trapezoidal equation corresponds to the trapezoidal matrix. In other words, any system of linear equations can be transformed into a trapezoidal matrix through the elementary row transformation of its augmented matrix, and the solution can be obtained.
The characteristics of trapezoidal matrix: all the elements in the lower left are zero, and the first non-zero element in each row is called the main element of the row.
Summarize the concrete solution results of different linear equations (unique solution, no solution, infinite solution), and then through strict proof, we can get the discriminant theorem about the solution of linear equations: firstly, the equations are transformed into trapezoid by elementary transformation, if the 0=d term appears in the obtained trapezoidal equations, the equations have no solution, if the 0=d term does not appear, the equations have a solution; When the equation has a solution, if the non-zero row number r of the trapezoid is equal to the unknown number n, the equation has a unique solution. If R gets a ladder through elementary transformation, the simplest form can be further obtained. The simplest form is characterized by the fact that all the elements above the principal component are zero, which makes it easier to solve the unknown value, but at the cost of more elementary transformation before. In the process of solving, the choice of ladder shape or the simplest shape depends on personal habits.
Linear equations with all zero constants are called homogeneous equations, and homogeneous equations must have zero solutions.
If the number of homogeneous equations is less than the number of unknowns, then the equations must have non-zero solutions.
Using the discriminant theorem of Gaussian elimination and solution, we can answer the above-mentioned (1) basic questions about the existence of solution and (2) how to solve it, which is the most basic theory based on linear equations.
For the special case of n equations with n unknowns, we find that the solution can be expressed by a combination of some coefficients, which is called the determinant of a linear equation (or matrix) according to certain rules. The characteristics of determinant: there is n! Items, the symbol of each item is determined by the inverse serial number of the angular scale arrangement, which is a number.
Through the study of determinant, some properties of determinant are obtained (such as exchanging the signs of two rows, there are two rows corresponding to a proportional value of zero, expanding by rows, etc.). ), which helps us to calculate determinant more conveniently.
Coefficient determinant can be used to judge the solution of n-ary linear equations of n equations, which is Cramer's rule.
In a word, determinant can be regarded as a part of the content derived from studying the special case that the number of equations is equal to the number of unknowns.