Elementary Linear Algebra

Transcription

1 Elementary Linear Algebra

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3 Linear algebra is the study of; linear sets of equations and their transformation properties. Linear algebra allows the analysis of; rotations in space, least squares fitting, solution of coupled differential equations, determination of a circle passing through three given points, as well as many other problems in mathematics, physics, and engineering.

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6 Chapter 1 Systems of Linear Equations and Matrices 1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix Operations 1.4 Inverses: Algebraic Properties of Matrices 1.5 Elementary Matrices and a Method for finding A More on Linear Systems and Invertible Matrices 1.7 Diagonal, Triangular, and Symmetric Matrices 1.8 Applications of Linear Systems 1.9 Leontief Input-Output Models

7 A linear equation in the variables x 1 x n is an equation that can be written in the form b and the coefficients a 1..a n are real or complex numbers, Usually known in advance. The subscript n may be any positive integer. In textbook examples and exercises, n is normally between 2 and 5. In real-life problems, n might be 50 or 5000, or even larger.

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12 Linear Systems in Two Unknowns

13 A system of linear equations has 1. no solution, or 2. exactly one solution, or 3. infinitely many solutions.

14 The Equation of a Plane What is x=4 in 2D and 3D? What is x=2 and y=2 in 2D and 3D? Find the equation of the plane passing through A(2,0,0) B(3,0,0) C(4,0,0)

15 Linear Systems in Three Unknowns

16 Matrix Notation The essential information of a linear system can be recorded compactly in a rectangular array called a matrix

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18 Solving a Linear System Describe an algorithm, or a systematic procedure, for solving linear systems. The basic strategy is to replace one system with an equivalent system (i.e., one with the same solution set) that is easier to solve

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25 Elementary Row Operations 1. Multiply a row through by a nonzero constant. 2. Interchange two rows. 3. Add a constant times one row to another

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31 Section 1.2 Gaussian Elimination Row Echelon Form Reduced Row Echelon Form: Achieved by Gauss Jordan Elimination

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34 We can find all the variables. So a solution exists; the system is consistent. So the solution is unique.)

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40 Homogeneous Systems All equations are set = 0 Theorem If a homogeneous linear system has n unknowns, and if the reduced row echelon form of its augmented matrix has r nonzero rows, then the system has n r free variables Theorem A homogeneous linear system with more unknowns than equations has infinitely many solutions

41 Matrices and Matrix Operations Definition 1 A matrix is a rectangular array of numbers. The numbers in the array are called the entries of the matrix. The size of a matrix M is written in terms of the number of its rows x the number of its columns. A 2x3 matrix has 2 rows and 3 columns

42 Arithmetic of Matrices A + B: add the corresponding entries of A and B A B: subtract the corresponding entries of B from those of A Matrices A and B must be of the same size to be added or subtracted ca (scalar multiplication): multiply each entry of A by the constant c

43 Multiplication of Matrices

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45 Diagonal, Triangular and Symmetric Matrices

46 Transpose of a Matrix AT

47 Ai j T A ji

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49 Transpose Matrix Properties

50 Trace of a matrix

51 Algebraic Properties of Matrices

52 Find if AB = BA

53 The identity matrix and Inverse Matrices

54 Inverse of a 2x2 matrix

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58 More on Invertible Matrices

59 Using Row Operations to find A -1 Begin with: Use successive row operations to produce:

60 Linear Systems and Invertible Matrices A. x = B A -1. A. x = A -1. B

61 A. x = B A -1. A. x = A -1. B x = A -1. B

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63 X 3 free variable If X 3 = t X 1 = -4/3 t X 2 = 2

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65 Applications of Linear Systems The concept of a network appears in a variety of applications. A network is a set of branches through which something flows. The branches might be: electrical wires through which electricity flows, pipes through which water or oil flows, traffic lanes through which vehicular traffic flows, economic linkages through which money flows,

66 In most networks, the branches meet at points, called nodes or junctions, where the flow divides We will restrict our attention to networks in which there is flow conservation at each node, by which we mean that the rate of flow into any node is equal to the rate of flow out of that node. This ensures that the flow medium does not build up at the nodes and block the free movement of the medium through the network.

67 Applications of Linear Systems

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74 Leontief Input-Output Models In 1973 the economist Wassily Leontief was awarded the Nobel prize for his work on economic modeling in which he used matrix methods to study the relationships between different sectors in an economy

75 A Homogeneous System in Economics Leontief input output (or production ) model

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79 The equilibrium price vector for the economy has the form

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81 0.2 Pc Pf Pm = Pc 0.3 Pc Pf Pm = Pf 0.5 Pc Pf Pm = Pm

82 0.8 Pc Pf Pm = Pc Pf Pm = Pc Pf Pm = Pm