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## CHAPTER 1 Linear Equations in Linear Algebra

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**§1.1 Systems of Linear Equations**• Basic concept • linear equation(线性方程) • system of linear equations(线性方程组), and its solution • Matrix(矩阵)**1.1.1 what is a linear equation?**Definition 1 (linear equation(线性方程)). A linear equation in the variables x1,…,xn is an equation of the form a1x1+ a2x2+ . . . + anxn = b (1) where b and the coefficients a1,…,an are real or complex numbers. eg.**What Is System of Linear Equations?**Definition 2 (system of linear equations(线性方程组)). A system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables- x1,…, xn a1,1x1+ a1,2x2+ . . . + a1,nxn = b1 a2,1x1+ a2,2x2+ . . . + a2,nxn = b2 （2） . . . am,1x1+ am,2x2+ . . . + am,nxn = bm**1.1.2 Solution of System of Linear Equations**Definition 3 (solution(解))). A list (S1, S2,… Sn) of numbers is called a solution of (2) iff (i.e. if and only if) all the equations in (2) are satisfied by substituting S1, S2,… Sn for X1, X2,… Xn. The set of all solutions of (2) is called the solution set (解集) of (2). Two systems of linear equations are said to be equivalent (等价) if they have the same solution set.**1.1.2 Solution of System of Linear Equations**• A system of linear equations has either 1. No solution, or 2. Exactly one solution, or • Infinitely many solutions. • Definition 4 (consistence (相容)). A system of linear equations is said • to be consistent if its solution set is nonempty (i.e. either one solution or infinitely many solutions), otherwise it is inconsistent. inconsistent ｝ consistent**1.1.2 Solution of System of Linear Equations**Fig(a). Exactly one solution Fig(b). no solutionFig(c). Infinitely many solutions**1.1.3 Matrix Notation P.4**Matrix Notation Coefficient matrix augmented matrix • Thesizeof a Matrix: how many rows and columns it has.**1.1.3 Matrix Notation**Definition 5 (matrix (矩阵)). A table of numbers with m rows (行) and n columns (列) as above is called an m n matrix. we normally use a capital letter such as A, B, X etc. to denote a matrix. Coefficient matrix augmented matrix**1.1.4 Solving a Linear System P.5**• Basic strategy (基本策略). • To replace one system with anequivalentsystem (one with the same solution set) that is easier to solve • Three basic operations (elementary operation(初等变换) to simplify a linear system 1. replace(倍加变换) one equation by the sum of itself and a multiple of another equation 2. interchange(交换) two equations 3. Scaling(倍乘变换) all the terms in an equation by a nonzero constant**Solving a Linear System ：**Sol: 4*[eq.1]+[eq.3] (½)*[eq.2] 4*[eq.3]+[eq.2] 。。。 3*[eq.2]+[eq.3] -1*[eq.3]+[eq.1] Upper triangular Back subsitution**augmented matrix**Sol:**Definition 6 (Row Equivalence (行等价)).**If matrix A can be transformed into matrix B by applying a series of elementary row operations on A , then we say A is row equivalent to B and denote this equivalence by A~ B. (P.7) If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set. (P.8)**1.1.5 Existence and Uniqueness Questions P.8**• 线性方程组解的存在和唯一性 • Two fundamental questions about a linear system 1. Is the system consistent; that is, does at least one solution exist? 2. If a solution exists, is it the only one; that is, is the solution unique?**Eg: Determine if the following system is consistent**Sol: From example1, we have We know x3, and substitute the value of x3 into eq.2 could get x2 , then could determine x1 from eq.1. So a solution exists; the system is consistent.**Eg:Determine if the following system is consistent:**Sol: The equation 0x1+0x2+0x3=(5/2) is never true, so the system is inconsistent.**§1.2 Row Reduction and Echelon Forms P.14**Basic concept: leading entry (先导元素) (row) echelon form (行阶梯形) echelon matrix (阶梯形矩阵) reduced (row) echelon form (简化阶梯形), reduced (row) echelon matrix(简化阶梯形矩阵) pivot position (主元位置)**1.2.1 Echelon Forms阶梯形 P.14***Definition 1:leading entry(先导元素): the first nonzero entry in a nonzero row. • Definition 2： A rectangular matrix is in echelon form (or row echelon form) if : 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros.**The following matrices are in echelon form(upper triangular**matrix):**reduced echelon form (or row reduced echelon form)**• Definition 3： A rectangular matrix is in reduced echelon form (or row reduced echelon form ,RREF) if : 1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros. 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column.**Theorem 1 : Uniqueness of the Reduced Echelon Form (p.15)**Each matrix is row equivalent to one and only one reduced echelon matrix. If a matrix A is row equivalent to an echelon matrix U, we call U an echelon form of A; If U is in reduced echelon form, we call Uthe reduced echelon form of A.**1.2.2 Pivot position(主元位置) P.15**• pivot: A pivot in a row echelon matrix U is a leading nonzero entry in a nonzero row. • Definition 4 pivot position: a position of a leading entry in an echelon form of the matrix. (P.16) • pivot column: a column that contains a pivot position.**Sol:**Example 2: Row reduce the matrix A below to echelon form, and locate the pivot columns of A. Interchange row1 and row4 Adding multiples of the first rows below:**Adding -5/2 times row 2 to row3, and add 3/2 times row 2 to**row 4 interchange rows 3 and 4 Note： There is no more than one in any row. There is no more than one in any colomn.**1.2.3 The Row Reduction Algorithm(行化简算法) P.17**Why? The reduced echelon form of a matrix A has the same solution as the original one. More, the reduced echelon form is easy for computing. Step1 Begin with the leftmost nonzero column. Step2 Select a nonzero entry in the pivot column as a pivot. Step3 Use row replacement operations to create zeros in all positions below the pivot. Step4 Apply steps 1-3 to the submatrix that remains. Repeat the process until there are no more nonzero rows to modify. Step5 Beginning with the rightmost pivot and working upward and to the left, create zeros above each pivot**Example 3: Transform the following matrix into reduced**echelon: Sol: Step1: Step2: Step3:**Step4:**(1) (2) Step5: (1) (2) (3) (4) The combination of steps 1-4 is called the forward phase of the row reductions algorithm. Steps 5 is called backward phase.**1.2.4. Solution of Linear Systems P.20**augmented matrix Associated system of equation (4) solution (5) • Basic variable(基本变量): any variable that corresponds to a pivot column in the augmented matrix of a system. • free variable(自由变量)：all nonbasic variables.**Example 4: Find the general solution(通解) of the**following linear system Sol:**The associated system now is**The general solution is: (7)**1.2.5 Parametric Descriptions of Solution Sets P.22**Solving a system amounts to finding a parametric description of the solution set or determine that the solution set is empty. The solution has many parametric descriptions. We make the arbitrary convention of always using the free variables as the parameters for describing a solution set.**1.2.5 Parametric Descriptions of Solution Sets P.22**• Back-Substitution • A computer program would solve system by back-substitution**Existence and Uniqueness Questions**Theorem 2 Existence and Uniqueness Theorem A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column– that is , if and only if an echelon form of the augmented matrix has no row of the form If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variable, or (ii) infinitely many solutions, when there is at least one free variable**Using Row Reduction to Solve A Linear System**1:Write the augmented matrix of the system. 2: Use the row reduction algorithm to obtain an equivalent augmented matrix in echelon form. If the system is inconsistent, Stop. 3: Continue row reduction to obtain the reduced echelon form. 4: Write the system of equations corresponding to the matrix obtained in step3. 5: Rewrite each nonzero equation form step4 so that its one basic variable is expressed in terms of any free variables appearing in the equation.**§1.3 Vector Equations P.28**Basic concept: column vector (列向量), linear combination (线性组合), Span (张)**§1.3 Vector Equations P.28**Vectors in R2 Geometric Description of R2 Vectors in R3 Vectors in Rn Linear Combination A Geometric Description of Span{v} and Span{u,v} Linear Combinations in Applications**1.3.1 Vector P.28**Definition 1 (vectors, 向量) A matrix with only one column is called a column vector, or simply a vector. • 1.3.1 Vectors in R2 A two-dimensional vector is a pair of numbers, surrounded by brackets(括号).**§1.3 Vector Equations**Vectors in R2 Notation: Different people use different notation for vector. v (boldface), (use arrows)**Vectors in R2**vectors are equal: If and only if they have the same corresponding entries. eg: Vector Addition: We add vectors in the obvious way, componentwise ≠ =**Scalar Multiplication(标量乘法) :**Notes: the vector cv has the same direction as v if c > 0 ,and the direction opposite to v if c < 0. Geometric Description of R2 Vector as points Vectors with arrows