System of Linear Equations
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1 Chapter 7 - S&B
2 Gaussian and Gauss-Jordan Elimination We will study systems of linear equations by describing techniques for solving such systems. The preferred solution technique- Gaussian elimination- answers the fundamental questions about a given linear system: does a solution exist, and if so, how many solutions are there? In general a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.. =. a m1 x 1 + a m2 x a mn x n = b m In this system, the a ij s and b i s are given real numbers; a ij is the coeffi cient of the unknown x j in the ith equation. A solution of a system is an n-tuple of real numbers x 1, x 2,.,x n, which satisfies each of the m equations.
3 Gaussian and Gauss-Jordan Elimination We are interested in the following three questions: 1 Does a solution exist? 2 How many solutions are there? 3 Is there an effi cient algorithm that computes actual solutions? There are three ways of solving such systems: (1) Substitution, (2) elimination of variables, and (3) matrix methods.
4 Substitution Solve one equation of system for one variable, say x n in terms of the other variables in that equation. Substitute this expression for x n into the other m 1 equations. The result is a new system of m 1 equations in the n 1unknowns x 1,..., x n 1 Continue this process by solving one equation in the new system for x n 1 and substituting this expression into the other m 2 equations to obtain a system of m 2 equations in the n 2 variables x 1,..., x n 2.
5 Substitution x 1 = 0.4x x x 2 = 0.2x x x x 3 = 0.5x x x
6 Elimination of variables x 1 0.4x 2 0.3x 3 = x x x 3 = x x x 3 = 95 We will also have to use the method back substitution. This procedure is called Gaussian elimination.
7 Elementary Row Operations Coeffi cient Matrix and Augmented Matrix Example x 1 2x 2 = 8 3x 1 + x 2 = 3 ( ) 1 2 and 3 1 ( ) Our three elementary equation operations now become elementary row operations: Interchange two rows of a matrix. Change a row by adding to it a multiple of another row, and Multiply each element in a row by the same non-zero number.
8 Elementary Row Operations Definition. A row of a matrix is said to have k leading zeros if the first k elements of the row are all zeros and the (k + 1)th element of the row is not zero. With this terminology, a matrix is in row echelon form if each row has more leading zeros than the row preceding it. Example ( ) and ( ) ( ) ( and )
9 Elementary Row Operations Identity Matrix I = Zero Matrix 0 =
10 Elementary Row Operations Pivot. It is the first nonzero entry in each row of a matrix in row echelon form Let us analyze the following example Definition. A row echelon matrix in which each pivot is a I and in which each column containing a pivot contains no other nonzero entries is said to be in reduced row echelon form.
11 System with many or no solutions In general two lines in the plane will be nonparallel and will cross in exactly one point. However, they can be parallel to each other. In this case, they will either coincide or they will never cross. lf they coincide; every point on either line is a solution (infinite many solutions). Let us analyze the following example p 1 + 2p 2 + 3p 3 = 1 3p 1 + 2p 2 + p 3 = 1
12 Rank - The Fundamental Criterion In order to answer the questions related to existence and uniqueness of solutions we need to analyze the rank. Definition.The rank of a matrix is the number of nonzero rows in its row echelon form. ( ) Fact 7.1. Let A and  be the coeffi cient matrix and augmented matrix respectively of a system of linear equations. Then rank A rank  rank A number of rows of A rank A number of columns of A
13 Fact 7.2. A system of linear equations with coeffi cient matrix A and augmented matrix  has a solution if and only if rank  = rank A Fact 7.3. A linear system of equations must have either no solutions, one solution, or infinitely many solutions. Fact 7.4. A system with a unique solution must have at least as many equations as unknowns. Homogeneous system (they have at least one solution) a 11 x a 1n x n = 0 a 21 x a 2n x n = 0... =. a m1 x a mn x n = 0
14 Fact 7.6. A homogeneous system of linear equations which has more unknows than equations must have infinitely many distinct solutions Fact 7.7. A system of linear equations with coeffi cient matrix A will have a solution for every choice of RHS b 1,.., b n if and only if rank A = number of rows of A Fact 7.9. Any system of linear equations having A as its coeffi cient matrix will have at most one solution for every choice of RHS b1,.., bm if and only if rank A = number of columns of A Fact A coeffi cient matrix A is nonsingular, that is, the corresponding linear system has one and only solution for every choice of RHS is number of rows of A =number of columns of A = rank A
15 Rank - The Fundamental Criterion Fact Consider the linear system of equations Ax = b If the number of equations < the number of unknowns, then: Ax = 0 has infinitely many solutions, for any given b, Ax = b has 0 or infinitely many solutions, and if rank A = number of equations, Ax = b has infinitely many solutions for every RHS b. If the number of equations = the number of unknowns, then: Ax = 0 has one or infinitely many solutions. for any given b, Ax = b has 0, 1, or infinitely many solutions, and if rank A = number of unknowns = number of equations, Ax = b has exactly 1 solution for every RHS b.
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