1. 1 SYSTEM OF LINEAR EQUATIONS BASE OF VECTOR SPACE

2. 2 System of linear equations System with m linear equations and n variables can be written as a 11 x 1 + a 12 x 2 + … + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + … + a 2n x n = b 2 : a m1 x 1 + a m2 x 2 + … + a mn x n = b m, – where the numbers a ij are the real coefficients of the system – x 1,..., x n are the real unknowns - variables – b 1,..., b m are the real values of right hand sides

3. 3 System of linear equations Examples

4. 4 Vector notation a 1, a 2, …, a n – a i is column vector of variable coefficients x 1, x 2, …, x n – x i is variable b is right hand side vector Linear combination of vectors x 1.a 1 + x 2.a 2 + … + x n.a n = b

5. 5 Matrix notation We can separate the coefficients a ij in a matrix as follows: Ax = b, where A is an m-by-n matrix, x is a column vector with n entries and b is a column vector with m entries.

6. 6 Solution - intuitively Simple approach Ax = b A -1 Ax = A -1 b Ex = A -1 b

7. 7 Gauss–Jordan elimination An algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix

8. 8 Historical notes The earliest presentation – The Nine Chapters on the Mathematical Art – Chinese mathematical text dated approximately 150 B.C.E Carl Friedrich Gauss (1777 – 1855) Wilhelm Jordan (1842–1899)

9. 9 Equivalent system of linear equations Two systems of linear equations are equivalent if they have the same solution. Intuitively 3x 1 + 6x 2 = 10 or 6x 1 + 12x 2 = 20

10. 10 Equivalent system of linear equations Using elementary row operations we receive equivalent systems Elementary row (column) operation – Multiply or divide row by a scalar – Add or subtract a scalar multiple of one row to another row – Switch the rows

11. 11 Solution of examples

12. 12 Gauss-Jordanian elimination Choosing of leading element – pivot on the main diagonal Replacing it by 1 (dividing row by the pivot) Replacing other elements in leading column by 0 (adding or subtracting leading row multiplied by proper number to actual row) Application of elementary row operations

13. 13 Row echelon form Leading 1 - The left–most nonzero element in each nonzero row is a 1. The leading 1 of each nonzero row is to the left of the leading 1 of any lower row. Only 0 are in column under leading 1 Any zero rows are at the bottom of the matrix.

14. 14 Reduced row echelon form Leading 1 - The left–most nonzero element in each nonzero row is a 1. The leading 1 of each nonzero row is to the left of the leading 1 of any lower row. All other elements in the column containing leading 1 are 0. Any zero rows are at the bottom of the matrix. Canonical form

15. 15 Solving of system of linear equations Augmented matrix of system – A  b – matrix A of variables coefficients with column vector b of right-hand side values. Transformation of this matrix A  b to reduced row echelon form by GJE

16. 16 Gauss-Jordan elimination Leading row Leading column Leading element PIVOT Actual row

17. 17 The computational complexity This algorithm can be used for systems with thousands of equations and unknowns. But the iterative methods are generally preferred for larger systems. Because GJE is numerically unstable, at least on pathological examples and may result in solutions far from the correct solution.

18. 18 Rank of matrix The rank of matrix is number of linearly independent rows (columns) in matrix. The rank of matrix can be set as number of nonzero rows (columns) in row echelon form of matrix.

19. 19 Frobenius theorem System of linear equations has at least one solution if and only if the rank of matrix of system is equal to the rank of augmented matrix of system.

20. 20 Solubility of system of linear equations The square system has a single solution when the rank of augmented matrix of system is the same as the rank of matrix of system. The system with more variables than equations has a infinite number of solution when the rank of augmented matrix of system is the same as the rank of matrix of system.

21. 21 Solubility of system of linear equations Three cases are possible for any given system of linear equations: The system has no solution (in this case, we say that the system is overdetermined) The system has a single solution (the system is exactly determined) The system has infinitely many solutions (the system is underdetermined).

22. 22 Basic solution Basic variables – Variables corresponding to columns of leading 1 (canonical basis) are equal to proper values in right-hand side vector. Nonbasic variables – Variables corresponding to other columns are equal to zero

23. 23 Nonbasic solution Nonbasic variables – Variables corresponding to nonbasic columns are equal to some (nonzero) values Basic variables – Values of variables corresponding to columns of leading 1 are computed according to values of RHS and values of nonbasic variables

24. 24 Parametric solution Nonbasic variables – Variables corresponding to nonbasic columns are parameters Basic variables – Values of variables corresponding to columns of leading 1 are expressed by RHS and parameters.

25. 25 Solution of examples

26. 26 Solution of examples

27. 27 Homogenous system of linear equations A system of the form Ax = 0 is called a homogenous system of linear equations. This system has trivial (zero) solution. The set of all solutions of such homogeneous system is called the null space of the matrix A – parametric solution

28. 28 Solution of examples 1210 4400 0110 1210 0-4 0 0110 100 0110 0000