2. Systems of Linear Equations.

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1 2. Systems of Linear Equations

2 2.1. Introduction to Systems of Linear Equations Linear Systems In general, we define a linear equation in the n variables x 1, x 2,, x n to be one that can be expressed in the form where a 1, a 2,, a n and b are constants and the a s are not all zero. In the special case where b=0, the equation has the form which is called a homogeneous linear equation.

3 2.1. Introduction to Systems of Linear Equations Linear Systems Example 1

4 2.1. Introduction to Systems of Linear Equations Linear Systems A finite set of linear equations is called a system of linear equations or a linear system. The variables in a linear system are called the unknowns. m equations, n unknowns a ij : i-th equation, j-th unknown Solution, solution set

5 2.1. Introduction to Systems of Linear Equations Linear Systems With Two and Three Unknowns Linear systems in two unknowns arise in connection with intersections of lines in R 2. A linear system is consistent if it has at least one solution and inconsistent if it has no solutions. Thus, a consistent linear system of two equations in two unknowns has either one solution or infinitely many solutions.

6 2.1. Introduction to Systems of Linear Equations Linear Systems With Two and Three Unknowns

7 2.1. Introduction to Systems of Linear Equations Linear Systems With Two and Three Unknowns Example 2 Example 4 Example 3 Example 5

8 2.1. Introduction to Systems of Linear Equations Augmented Matrices And Elementary Row Operations Augmented matrix

9 2.1. Introduction to Systems of Linear Equations Augmented Matrices And Elementary Row Operations The succession of simpler systems can be obtained by eliminating unknowns systematically using three types of operations: 1. Multiply an equation through by a nonzero constant. 2. Interchange two equations. 3. Add a multiple of one equation to another. Elementary row operations

10 2.1. Introduction to Systems of Linear Equations Augmented Matrices And Elementary Row Operations Example 6

11 2.1. Introduction to Systems of Linear Equations Augmented Matrices And Elementary Row Operations Example 6 Determine whether the vector w=(9,1,0) can be expressed as a linear combination of the vectors

12 2.2. Solving Linear Systems by Row Reduction Echelon Forms Reduced row echelon form 1. If a row does not consist entirely of zeros, then the first nonzero number in the row is a 1. We call this a leading If there are any rows that consist entirely of zeros, then they are grouped together at the bottom of the matrix. 3. In any two successive rows that do not consist entirely of zeros, the leading 1 in the lower row occurs farther to the right than the leading 1 in the higher row. 4. Each column that contains a leading 1 has zeros everywhere else. A matrix that has the first three properties is said to be in row echelon form.

13 2.2. Solving Linear Systems by Row Reduction Echelon Forms Example 1

14 2.2. Solving Linear Systems by Row Reduction Echelon Forms Example 2 A matrix in row echelon form has zeros below each leading 1, whereas a matrix in reduced echelon form has zeros below and above each leading 1.

15 2.2. Solving Linear Systems by Row Reduction Echelon Forms If by a sequence of elementary row operations, the augmented matrix for a system of linear equations is put in reduced row echelon form, then the solution set can be obtained either by inspection, or by converting certain linear equations to parametric form. Example 3

16 2.2. Solving Linear Systems by Row Reduction Echelon Forms Example 4

17 2.2. Solving Linear Systems by Row Reduction General Solutions As Linear Combinations of Column Vectors For many purposes, it is desirable to express a general solution of a linear system as a linear combination of column vectors.

18 2.2. Solving Linear Systems by Row Reduction Gauss-Jordan And Gaussian Elimination A step-by-step procedure that can be used to reduce any matrix to reduced row echelon form by elementary row operations. Forward phase, backward phase Gaussian-Jordan elimination Gaussian elimination

19 2.2. Solving Linear Systems by Row Reduction Some Facts About Echelon Forms 1. Every matrix has a unique reduced row echelon form; that is, regardless of whether one uses Gaussian-Jordan elimination or some other sequence of elementary row operations, the same reduced row echelon form will result in the end. 2. Row echelon forms are not unique; that is, different sequences of elementary row operations may result in different row echelon forms for a given matrix. However, all of the row echelon forms have their leading 1 s in the same positions and all have the same number of zero rows at the bottom. The positions that have the leading 1 s are called the pivot positions in the augmented matrix, and the columns that contain the leading 1 s are called pivot columns.

20 2.2. Solving Linear Systems by Row Reduction Some Facts About Echelon Forms Example 5

21 2.2. Solving Linear Systems by Row Reduction Back Substitution Example 6 Back substitution

22 2.2. Solving Linear Systems by Row Reduction Homogeneous Linear Systems A linear equation is said to be homogeneous if its constant term is zero. A linear system is homogeneous if each of its equations is homogeneous.

23 2.2. Solving Linear Systems by Row Reduction Homogeneous Linear Systems Observe that every homogeneous linear system is consistent, since is a solution. This is called the trivial solution. All other solutions, if any, are called nontrivial solutions. If the homogeneous linear solution has some nontrivial solution Then it must have infinitely many solutions, since is also a solution for any scalar t.

24 2.2. Solving Linear Systems by Row Reduction Homogeneous Linear Systems Example 7

25 2.2. Solving Linear Systems by Row Reduction The Dimension Theorem for Homogeneous Linear Systems REMARK It is important to keep in mind that this theorem is only applicable to homogeneous linear systems. Indeed, there exist nonhomogeneous linear systems with more unknowns than equations that have no solutions.

26 2.3. Applications of Linear Systems Global Positioning

27 2.3. Applications of Linear Systems Global Positioning Example 1

28 2.3. Applications of Linear Systems Global Positioning Example 1 The quadratic terms in all of these equations are the same, so if we subtract each of the last three equations from the first one, we obtain the linear system

29 2.3. Applications of Linear Systems Global Positioning Example 1 To find s we can substitute these expressions into any of the four quadratic equations from the satellite.

30 2.3. Applications of Linear Systems Network Analysis Loosely stated, a network is a set of branches through which something flows. The branches meet at points, called nodes or junctions, where the flow divides. Three basic properties: 1. One-directional flow: At any instant, the flow in a branch is in one direction only. 2. Flow conservation at a node: the rate of flow into a node is equal to the rate of flow out of the node. 3. Flow conservation in the network: the rate of flow into the network is equal to the rate of flow out of the network.

31 2.3. Applications of Linear Systems Network Analysis Example 2 Figure 2.3.3a shows a network in which the flow rate and direction of flow in certain branches are known. Find the flow rates and directions of flow in the remaining branches.

32 2.3. Applications of Linear Systems Network Analysis Example 3 (a) How many vehicles per hour should the traffic light let through to ensure that the average number of vehicles per hour flowing into the complex is the same as the average number of vehicles flowing out? (b) Assuming that the traffic light has been set to balance the total flow in and out of the complex, what can you say about the average number of vehicles per hour that will flow along the streets that border the complex?

33 2.3. Applications of Linear Systems Electrical Circuits

34 2.3. Applications of Linear Systems Electrical Circuits Example 4 Determine the current I in the circuit shown in Figure

35 2.3. Applications of Linear Systems Electrical Circuits Example 5 Determine the current I 1, I 2, and I 3 in the circuit shown in Figure I 1 =6A, I 2 =-5A, and I 3 =1A

36 2.3. Applications of Linear Systems Balancing Chemical Equations Chemical formulas Chemical equation reactants products balanced

37 2.3. Applications of Linear Systems Balancing Chemical Equations x 1 =1, x 2 =2, x 3 =1, x 4 =2

38 2.3. Applications of Linear Systems Balancing Chemical Equations Example 6 Balance the chemical equation

39 2.3. Applications of Linear Systems Polynomial Interpolation Polynomial interpolation: finding a polynomial whose graph passes through a specified set of points in the plane.

40 2.3. Applications of Linear Systems Polynomial Interpolation

41 2.3. Applications of Linear Systems Polynomial Interpolation Example 7 Find a cubic polynomial whose graph passes through the points

42 2.3. Applications of Linear Systems Polynomial Interpolation Example 8 Approximate integration

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