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Chapter 1 – Linear Equations

Outline1.1 Introduction to Linear Systems1.2 Matrices, Vectors, and Guass-Jordan Elimination1.3 On the Solutions of Linear Systems

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1.1 Introduction to Linear Systems

• Solving systems of linear equations– For small systems

• Ad hoc methods

– For larger systems• Require more systematic methods

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Introduction to Linear Systems (II)

• Nine Chapters on the Mathematical Art (數書九章 ) (2000 years ago in Chinese text, Chiu-chang Suan-shu):– The yield of one sheaf of inferior grain, two sheaves of medium grain,

and three sheaves of superior grain is 39 tou. The yield of one sheaf of inferior grain, three sheaves of medium grain, and two sheaves of superior grain is 34 tou. The yield of three sheaves of inferior grain, two sheaves of medium grain, and one sheaf of superior grain is 26 tou. What is the yield of inferior, medium, and superior grain?

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Introduction to Linear Systems (III)

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• We can check that x=2.75, y=4.25, z=9.25

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Geometric Interpretation

• Each of the three equations of the system defines a plane in x-y-z space.

• Only one solution (a unique solution): (x, y, z)=(2.75, 4.25, 9.25)

• This means that the planes defined by the three equations intersect at the point (x, y, z)=(2.75, 4.25, 9.25).

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Geometric Interpretation (II)

• Infinitely many solutions and no solutions at all

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A System with Infinitely Many Solutions

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• The general solution(x, y, z)=(t+2, -2t-1, t)=(2, -1, 0)+t(1, -2, 1)

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A System without Solutions

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• This system is inconsistent for 0=-6.

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1.2 Matrices, Vectors, and Gauss-Jordan Elimination

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• To label the entries of a 3×4 matrix A with double subscripts:

• The entry aij is located in the ith row and the jth column.

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Matrices

• Two matrices A and B are equal if they have the same size and if corresponding entries are equal: aij = bij .

• If the number of rows of a matrix A equals the number of columns (A is n×n), then A is called a square matrix– The entries a11, a22, . . . , ann form the (main) diagonal of A.

– A square matrix A is called diagonal if all its entries off the main diagonal are zero; that is, aij = 0 whenever i ≠ j .

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Matrices (II)

• A square matrix A is called Upper triangular if all its entries below the main diagonal are zero; that is, aij = 0 whenever i exceeds j .

• Lower triangular matrices are defined analogously.• A matrix whose entries are all zero is called a zero

matrix and is denoted by 0 (regardless of its size).

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Matrices (III)

• The matrices B, C, D, and E are square, C is diagonal, C and D are upper triangular, and C and E are lower triangular.

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Vectors

• The standard representation of a vector in the Cartesian coordinate plane is as an arrow (a directed line segment) connecting the origin to the point (x,y), as shown in the following.

(x, y)

0

y

xv

y

xv

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Vectors (II)

• A matrix with only one column is called a column vector, or simply a vector.

• The entries of a vector are called its components.

• The set of all column vectors with n components is denoted by Rn.

• The n columns of an m × n matrix are vectors in Rm.

• is a vector in R4.

• is a row vector with 5 components

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Linear Equations and Matrices

• Consider the system

• The matrix contains the coefficients of the system, called its coefficient matrix.

• The matrix displays all the numerical information contained in the system, is called its augmented matrix.

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Linear Equations and Matrices (II)

• The solution is often represented as a vector:

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Gauss-Jordan Elimination

• Consider the system

• The augmented matrix of this system

• The cursor is placed at the top position of the first nonzero column of the matrix

• Our goal is to make the cursor entry equal to 1.

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Gauss-Jordan Elimination (Step 1)

• If the cursor entry is 0, swap the cursor row with some row below to make the cursor entry nonzero.

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Gauss-Jordan Elimination (Step 2)

• Divide the cursor row by the cursor entry to make the cursor entry equal to 1.

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Gauss-Jordan Elimination (Step 3)

• Eliminate all other entries in the cursor column by subtracting suitable multiples of the cursor row from the other rows.

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Gauss-Jordan Elimination (Step 4)

• Move the cursor down diagonally (i.e., down one row and over one column). If the new cursor entry and all entries below are zero, move the cursor to the next column (remaining in the same row). Repeat this step if necessary. Then return to step 1.

• When we try to apply step 4 to this matrix, we run out of columns; the process of row reduction comes to an end. We say that the matrix E is in reduced row-echelon form, or rref for short. We write E = rref(M), where M is the augmented matrix of the system.

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Reduced Row-Echelon Form

• A matrix is in reduced row-echelon form if it satisfies all of the following conditions:– If a row has nonzero entries, then the first nonzero entry is 1, called the

leading 1 in this row.

– If a column contains a leading 1, then all other entries in that column are zero.

– If a row contains a leading 1, then each row above contains a leading 1 further to the left.

• If the echelon form contains the equation 0=1, then there are no solutions; the system is inconsistent.

• The operations performed in Steps 1, 2, and 3 of Guass-Jordan elimination are called elementary row operations: Swap two rows, divide a row by a scalar, or subtract a multiple of a row from another row.

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1.3 On the Solutions of Linear Systems; Matrix Algebra

• The reduced row-echelon forms of the augmented matrices of three systems are given. How many solutions are there in each case?

–

0

1

0

0

000

000

100

021

0

2

1

000

100

021

3

2

1

100

010

001

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(Fact 1.3.1) Number of Solutions of a Linear System

• A system of equations is said to be consistent if there is at least one solution; it is inconsistent if there are no solutions.

• A linear system is inconsistent if (and only if) the reduced row-echelon form of its augmented matrix contains the row [0 0 … 0 | 1], representing the equation 0=1.

• If a linear system is consistent, then it has either– Infinitely many solutions (if there is at least one free variable), or

– Exactly one solution (if all the variables are leading).

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Rank

• (Definition 1.3.2) Rank: The rank of a matrix A is the number of leading 1’s in rref(A).

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• Consider a system of m linear equations with n unknowns. Its coefficient matrix A has the size m × n.– rank(A) ≤ m and rank(A) ≤ n.

– If rank(A) = m, then the system is consistent.

– If rank(A) = n, then the system has at most one solution.

– If rank(A) < n, then the system has either infinitely many solutions, or none.

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No Solutions, a Unique Solution, and Infinitely Many Solutions

• (Fact 1.3.3) A linear system with fewer equations than unknowns has either no solutions or infinitely many solutions.

• Consider two equations in three variables: Two planes in space either intersect in a line or are parallel.

• (Fact 1.3.4) A linear system of n equations with n unknowns has a unique solution if (and only if) the rank of its coefficient matrix A is n. This means that

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(Definition 1.3.5) Sums of Matrices & Scalar Multiples of Matrices

• (Sums of matrices) The sum of two matrices of the same size is defined entry by entry:

• (Scalar multiples of matrices) The product of a scalar with a matrix is defined entry by entry:

nmnmnn

mm

nmn

m

nmn

m

baba

baba

bb

bb

aa

aa

11

111111

1

111

1

111

nmn

m

nmn

m

kaka

kaka

aa

aa

k

1

111

1

111

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The Matrix Form of the Linear System

• (Definition 1.3.6)

• The product is defined only if the number of columns of A matches the number of components of .

• The product is a vector in Rm.

• The matrix form of the linear system for the vector form is bxA

x

bvxvx nn

11

xA

xA

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The Vector Form of a Linear System

• Consider the linear system

• The vector form of the linear system

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The Vector Form of a Linear System (II)

• Consider the general linear system

• The vector form of the linear system

• (Definition 1.3.7) Linear combinations

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(Fact 1.3.8) The Product Ax in Terms of The Rows of A

• If A is an n×m matrix with row vectors , andis a vector in Rm, then

xw

xw

x

w

w

xA

nn

11

nww

,,1 x

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(Fact 1.3.9) Algebraic Rules for Ax

• 2 important algebraic rules for the product

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(Fact 1.3.10) Matrix Form of a Linear System

• We can write the linear system with augmented matrix in matrix form as

• (Example 14) Write the system

in matrix form.

bA

bxA

8649

7532

321

321

xxx

xxx

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Matrices Operations

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