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2. Systems of Linear Equations2. Systems of Linear Equations
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2.1. Introduction to Systems of Linear Equations2.1. Introduction to Systems of Linear Equations
Linear Systems
In general, we define a linear equation in the n variables x1, x2, …, xn to be one that can be expressed in the form
where a1, a2, …, an 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.
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2.1. Introduction to Systems of Linear Equations2.1. Introduction to Systems of Linear Equations
Linear Systems
Example 1Example 1
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2.1. Introduction to Systems of Linear Equations2.1. 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
aij: i-th equation, j-th unknown
Solution, solution set
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2.1. Introduction to Systems of Linear Equations2.1. 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 R2.
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.
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2.1. Introduction to Systems of Linear Equations2.1. Introduction to Systems of Linear Equations
Linear Systems With Two and Three Unknowns
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2.1. Introduction to Systems of Linear Equations2.1. Introduction to Systems of Linear Equations
Linear Systems With Two and Three Unknowns
Example 2Example 2
Example 3Example 3
Example 4Example 4
Example 5Example 5
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2.1. Introduction to Systems of Linear Equations2.1. Introduction to Systems of Linear Equations
Augmented Matrices And Elementary Row Operations
Augmented matrix
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2.1. Introduction to Systems of Linear Equations2.1. 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
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2.1. Introduction to Systems of Linear Equations2.1. Introduction to Systems of Linear Equations
Augmented Matrices And Elementary Row Operations
Example 6Example 6
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2.1. Introduction to Systems of Linear Equations2.1. Introduction to Systems of Linear Equations
Augmented Matrices And Elementary Row Operations
Determine whether the vector w=(9,1,0) can be expressed as a linear combination of the vectors
Example 6Example 6
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2.2. Solving Linear Systems by Row Reduction2.2. 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 1.
2. 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.
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2.2. Solving Linear Systems by Row Reduction2.2. Solving Linear Systems by Row Reduction
Echelon Forms
Example 1Example 1
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2.2. Solving Linear Systems by Row Reduction2.2. Solving Linear Systems by Row Reduction
Echelon Forms
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.
Example 2Example 2
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2.2. Solving Linear Systems by Row Reduction2.2. 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 3Example 3
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2.2. Solving Linear Systems by Row Reduction2.2. Solving Linear Systems by Row Reduction
Echelon Forms
Example 4Example 4
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2.2. Solving Linear Systems by Row Reduction2.2. 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.
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2.2. Solving Linear Systems by Row Reduction2.2. 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
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2.2. Solving Linear Systems by Row Reduction2.2. 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.
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2.2. Solving Linear Systems by Row Reduction2.2. Solving Linear Systems by Row Reduction
Some Facts About Echelon Forms
Example 5Example 5
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2.2. Solving Linear Systems by Row Reduction2.2. Solving Linear Systems by Row Reduction
Back Substitution
Example 6Example 6
Back substitution
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2.2. Solving Linear Systems by Row Reduction2.2. 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.
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2.2. Solving Linear Systems by Row Reduction2.2. 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.
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2.2. Solving Linear Systems by Row Reduction2.2. Solving Linear Systems by Row Reduction
Homogeneous Linear Systems
Example 7Example 7
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2.2. Solving Linear Systems by Row Reduction2.2. 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.
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Global Positioning
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Global Positioning
Example 1Example 1
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Global Positioning
Example 1Example 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
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Global Positioning
Example 1Example 1
To find s we can substitute these expressions into any of the four quadratic equations from the satellite.
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2.3. Applications of Linear Systems2.3. 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.
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Network Analysis
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.
Example 2Example 2
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Network Analysis
(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?
Example 3Example 3
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Electrical Circuits
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Electrical Circuits
Determine the current I in the circuit shown in Figure 2.3.9.Example 4Example 4
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Electrical Circuits
Example 5Example 5Determine the current I1, I2, and I3 in the circuit shown in Figure 2.3.10.
I1=6A, I2=-5A, and I3=1A
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Balancing Chemical Equations
Chemical formulas
Chemical equation
reactants products
balanced
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Balancing Chemical Equations
x1=1, x2=2, x3=1, x4=2
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Balancing Chemical Equations
Balance the chemical equationExample 6Example 6
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Polynomial Interpolation
Polynomial interpolation: finding a polynomial whose graph passes through a specified set of points in the plane.
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Polynomial Interpolation
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Polynomial Interpolation
Find a cubic polynomial whose graph passes through the pointsExample 7Example 7
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2.3. Applications of Linear Systems2.3. Applications of Linear Systems
Polynomial Interpolation
Approximate integrationExample 8Example 8