Presentation on theme: "Chapter 1 Systems of Linear Equations and Matrices"— Presentation transcript:
1 Chapter 1 Systems of Linear Equations and Matrices Introduction to System of Linear EquationsGaussian EliminationMatrices and Matrix OperationsInverses; Rules of Matrix ArithmeticElementary Matrices and a Method for FindingFurther Results on Systems of Equations and InvertibilityDiagonal, Triangular, and Symmetric Matrices
2 1.1 Introduction to systems of linear equations Any straight line in xy-plane can be represented algebraically by an equation ofthe form:General form: Define a linear equation in the n variables :where a1, a2, …, an and b are real constants.The variables in a linear equation are sometimes called unknowns.
3 Linear EquationsExamples:Some linear equations:Some NON linear equations:Observe thatA linear equation does not involve any products or roots of variablesAll variables occur only to the first power and do not appear as arguments for trigonometric, logarithmic, or exponential functions.
4 Solutions of Linear Equations A solution of a linear equation is a sequence of n numbers s1, s2, …, sn such that the equation is satisfied.The set of all solutions of the equation is called its solution set or generalsolution of the equation.Example:(a) Find the solution of 3x+4y = 5
6 Linear SystemsA finite set of linear equations in the variables x1, x2, …, xn is called asystem of linear equations or a linear system.A sequence of numbers s1, s2, …, sn is called a solution of the system if every equation is satisfied.A system has no solution is said to be inconsistent.If there is at least one solution of the system, it is called consistent.Every system of linear equations has either no solutions, exactly one solution, or infinitely many solutions.A general system of two linear equations:Two line may be parallel – no solutionTwo line may be intersect at only one point – one solutionTwo line may coincide – infinitely many solutions
7 An arbitrary system of m linear equations in n unknown can be written as Where x1, x2, …, xn are unknowns and the subscripted a’s and b’s denoteConstants.Note that is in the ith equation and multiplies unknown
8 Augmented MatrixA system of m linear equations in n unknowns can be written as a rectangular arrayof numbers:This is called the augmented matrix for the system.Example:
9 Elementary Row Operations Since the rows of an augmented matrix correspond to the equations, we can apply the following three types of operations to solve systems of linear equations.Multiply an equation through by an nonzero constantInterchange two equationAdd a multiple of one equation to anotherThese operations are called elementary row operations.We will go to details in the next session.
10 1.2 Gaussian EliminationA matrix which has the following properties is in reduced row echelon Form.If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leader 1.If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix.In any two successive rows that do not consist entirely of zeros, the leader 1 in the lower row occurs farther to the right than the leader 1 in the higher row.Each column that contains a leader 1 has zeros everywhere else.A matrix that has the first three properties is said to be in row echelon form.Note: A matrix in reduced row-echelon form is of necessity in row echelonform, but not conversely
11 Row-Echelon and Reduced Row-Echelon Forms Example: Determine which of the following matrices are in row-echelon form, reducedRow-echelon form, both, or neither.Solution:
12 Example: Determine which of the following matrices are in row-echelon form, reduced Row-echelon form, both, or neither.Solution:
13 More on Row-Echelon and Reduced Row-Echelon Form All matrices of the following types are in row-echelon form (any real numbers substituted for the *’s. ) :All matrices of the following types are in reduced row-echelon form (anyreal numbers substituted for the *’s. ) :
14 Solutions of Linear Equations Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form. Solve the system.Solution (a):
15 Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form.Solve the system.Solution (b):
16 Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form.Solve the system.Solution (c):
17 Example: Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given reduced row-echelon form.Solve the system.Solution (d):
18 Elimination MethodElimination procedure to reduce a matrix to reduced row-echelon form.Step 1. Locate the leftmost column that does not consist entirely of zeros.Step 2. Interchange the top row with another row, if necessary, to bring a nonzeroentry to the top of the column found in step 1.R1R2
19 Step 3. If the entry that is now at the top of the column found in step 1 is a, multiply The first row by 1/a in order to introduce a leading 1.Step 4. Add suitable multiples of the top row to the rows below so that all entriesBelow the leading 1 become zeros.Step 5. Cover the top row in the matrix and begin again with Step 1 applied to theSubmatrix that remains. Continue in this way until the entire matrix is in row-echelonform.
20 Step 6. Beginning with the last nonzero row and working upward, add suitable multiples of each row to the rows above to introduce zeros above the leading 1’sStep1~Step5: the above procedure produces a row-echelon form andis called Gaussian eliminationStep1~Step6: the above procedure produces a reduced row-echelonform and is called Gaussian-Jordan elimination
21 Every matrix has a unique reduced row-echelon form but a row-echelon form of a given matrix is not uniqueExample: Solve by Gauss-Jordan eliminationSolution:
22 Homogeneous Linear Systems A system of linear equations is said to be homogeneous if the constantterms are all zero; that is, the system has the form:Every homogeneous system of linear equation is consistent, since all such system have x1 = 0, x2 = 0, …, xn = 0 as a solution. This solution is called the trivial solution. If there are another solutions, they are called nontrivial solutions.There are only two possibilities for its solutions:There is only the trivial solutionThere are infinitely many solutions in addition to the trivial solution
23 Example: Solve the homogeneous system of linear equations by Gauss-Jordan elimination Solution:
24 Theorem 1.2.1A homogeneous system of linear equations with more unknownsthan equations has infinitely many solutions.RemarkThis theorem applies only to homogeneous system!A nonhomogeneous system with more unknowns than equations need not be consistent; however, if the system is consistent, it will have infinitely many solutions.