Presentation on theme: "Chapter 1 – Linear Equations"— Presentation transcript:
1 Chapter 1 – Linear Equations Outline Introduction to Linear Systems Matrices, Vectors, and Guass-Jordan Elimination On the Solutions of Linear Systems
2 1.1 Introduction to Linear Systems Solving systems of linear equationsFor small systemsAd hoc methodsFor larger systemsRequire more systematic methods
3 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?
4 Introduction to Linear Systems (III) We can check that x=2.75, y=4.25, z=9.25
5 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).
6 Geometric Interpretation (II) Infinitely many solutions and no solutions at all
7 A System with Infinitely Many Solutions The general solution (x, y, z)=(t+2, -2t-1, t)=(2, -1, 0)+t(1, -2, 1)
8 A System without Solutions This system is inconsistent for 0=-6.
9 1.2 Matrices, Vectors, and Gauss-Jordan Elimination 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.
10 MatricesTwo 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 matrixThe 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 .
11 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).
12 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.
13 VectorsThe 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)
14 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
15 Linear Equations and Matrices Consider the systemThe 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.
16 Linear Equations and Matrices (II) The solution is often represented as a vector:
17 Gauss-Jordan Elimination Consider the systemThe augmented matrix of this systemThe cursor is placed at the top position of the first nonzero column of the matrixOur goal is to make the cursor entry equal to 1.
18 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.
19 Gauss-Jordan Elimination (Step 2) Divide the cursor row by the cursor entry to make the cursor entry equal to 1.
20 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.
21 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.
22 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.
23 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?
24 (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 eitherInfinitely many solutions (if there is at least one free variable), orExactly one solution (if all the variables are leading).
25 Rank(Definition 1.3.2) Rank: The rank of a matrix A is the number of leading 1’s in rref(A).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.
26 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
27 (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:
28 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
29 The Vector Form of a Linear System Consider the linear systemThe vector form of the linear system
30 The Vector Form of a Linear System (II) Consider the general linear systemThe vector form of the linear system(Definition 1.3.7) Linear combinations
31 (Fact 1.3.8) The Product Ax in Terms of The Rows of A If A is an n×m matrix with row vectors , and is a vector in Rm, then
32 (Fact 1.3.9) Algebraic Rules for Ax 2 important algebraic rules for the product
33 (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.