"Linear Algebra and its application"-Solution of linear equation Group

Solution of a linear equation group:

With the introduction of the general solution of the three equations of matrix equation, linear equation group and vector equation, we can express a linear equations in a relatively simple way. The following form is available:

Ax = b. Where A is a matrix of M x n, corresponding to the coefficient matrix of a linear equation group, and X is a vector of r^n, the n unknown amount is recorded, and B is the attempt to the right of the equations of linear equation, which is essentially a vector of r^n, then based on this expression, we begin to discuss the structure of the solution of the linear equations ( Similar to the differential equation we discuss its general solution, the special solution of a process).

We are starting from B, we are faced with the following two situations.

Homogeneous equation:

b = 0 (note that the 0 here represents the vector). We call such a linear equation group as the homogeneous equations.

This is where a linear equation group can be written in ax = 0 form. It is easy to see that this equation must have a solution x0, each of its components are 0, corresponding to the linear equation group, namely x1=0,x2=0,x3=0,... xn=0, where we call this solution x0 for the ordinary solution.

But this does not affect the solution set of the homogeneous equation, and an example is given to discuss the solution set of the homogeneous linear equations. (here involves a vector of the parameter expression form, very simple needless to say)

Non-homogeneous equation:

B≠0, we solve the value of b non-zero vector again based on the example given.

Comparing the answers to the two questions, we seem to have a surprise finding that in the solution of the homogeneous equation, we get the expression of the vector parameter in the form of x = SV (S is constant, V is the vector), and in the solution of the non-homogeneous equation, we get the expression of the vector parameter x = p + SV (S is the parameter P is a vector), in fact, we can do more experiments to test, found that for the homogeneous equation set ax = 0 and non-homogeneous equations of the solution set, there are the above-mentioned formal law, so we define the non-homogeneous equations of the solution set X = p + SV is the special solution of the equation, that is, the following theorem is established

Interesting thing, for the solution is r^2 or r^3 situation, we can add the base vector to describe the relationship between the two sets of solutions geometrically, that is, the non-homogeneous equation of the group of the arbitrary solution can be regarded as its special solution vector p along the corresponding homogeneous equation of arbitrary solution v translation any unit length obtained.

From the above example, it is not difficult to add some steps to integrate the solution into the form of the parameter vectors in solving the solution set of the linear equations.

The algorithm for the solution set of the compatible equations is written as a parameter vector:

"Linear Algebra and its application"-Solution of linear equation Group