[Line notes] Chapter One solution of linear equation Group

Chapter One solution of linear equation Group

• Algebra originates from Solution equations (algebraic equations)
  • One dollar, one yuan two times, one yuan three times, one yuan four times have to find the root formula (through the coefficients of the finite times add, subtract, multiply, divide, the powers, the root to get the solution), more than five times the equation will no longer have the roots formula (modern algebra)
  • Binary one-time equations, ternary first-time equations 、......、 N-element first-time equations (linear algebra study objects)
  • Advanced Algebra--linear algebra + polynomial theory

1. The same solution deformation, linear combination, elementary transformation, elimination method of linear equation group

• Example 1
  
  

1. The same solution deformation : with 3 kinds of the same solution deformation must be able to transform the equation set for the ladder type
   1. Swap two equation positions
   2. Multiply an equation by a number of non-0 C on either side
   3. To use the K-multiplier of one equation to another
2. linear combination : Set is a linear combination of some equations, called. (consisting of a group of equations and the same solution)
   • Example 2
     
     Because of this, the first equation is superfluous.
     
3. So so
   M equations of n unknowns, coefficients are the coefficients of the first unknowns of the first equation. (Number of fields), at this point the solution is also in.

• Number fields : Subsets of complex numbers are closed for addition, subtraction, multiplication, and divide (denominator not 0), called Number fields. Such as.

2. Related concepts of matrices

1. The above equations are determined entirely by the table,
   
   • A table consisting of the number of rows (), bounded by parentheses (or square brackets), called a matrix on a number of fields.
   • The lines in the matrix are called vectors ( row vectors ), as a vector, which can be thought of as a single line of matrices. Similarly, each column is called a column vector .
   • 0 Vectors :.
2. Elementary transformation of matrices : must be transformed from elementary to ladder-shaped matrix, called Matrix elimination method of Equation Group
   1. Swap two lines
   2. Multiply a row
   3. A row k doubly to another line

3. Matrix elimination method for solving linear equations

2. Consider the equations.
   
   Called the coefficient matrix , which is the augmented matrix .
   • The equation group and its augmented matrix are uniquely determined by each other.
   • For the elementary transformation, the ladder-shaped equation Group is then solved.
   • Cases
     
     Solution:
     
     Visible ladder shape can be irregular
     Rewritten as
     Let the value of liberty be the solution
     
     wherein, called the free unknown amount , the original equation of the infinite multi-group solution.
   • Proposition : When the augmented matrix of the equations is set to a ladder shape, with a non-0 line, and the last non-0 lines, the equations have a free unknown quantity, and thus have infinite group solutions. ( rank of matrix: not 0 lines after ladder)
   • theorem 1: After the augmented matrix is transformed into a ladder shape with the elementary row transformation, the rank of the coefficient matrix is recorded as the rank of the augmented matrix (yes), then
     A. No solution for the time equation group
     At this point, the last line is no solution, the last behavior in the ladder form
     The solution of the equation group
     B. The time Equation Group has the solution
     A. (number of unknowns), there is infinite set of solutions, at this time there is a free unknown quantity
     B. When there is only one set of solutions
4. General Solution : The set of equations have infinite groups of solutions, there is a free unknown quantity, make
   Which is called the General solution of the equation group.
   The general solution can also be written as a vector
6. Special Solution : A specific solution in the general solution.
8. Solution Set :

4. Homogeneous linear equations (the right constant is all 0)

Only one homogeneous equation is considered here.

1. The rank of the coefficient matrix must have a non-0 solution, when only 0 solutions
2. If the number of equations of the homogeneous linear equation Group (number of unknowns), there must be a non-0 solution. (There must be no 0 solutions at this time)

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[Line notes] Chapter One solution of linear equation Group