CHAPTER 1 Linear Equations in Linear Algebra. §1.1 Systems of Linear Equations Basic concept linear equation( 线性方程 ) system of linear equations( 线性方程.

CHAPTER 1 Linear Equations in Linear Algebra

1.1 Systems of Linear Equations Basic concept linear equation( ) system of linear equations( ), and its solution Matrix( )

1.1.1 what is a linear equation? Definition 1 (linear equation( )). A linear equation in the variables x 1,,x n is an equation of the form a 1 x 1 + a 2 x 2 +... + a n x n = b (1) where b and the coefficients a 1, ,a n are real or complex numbers. eg.

What Is System of Linear Equations? Definition 2 (system of linear equations( )). A system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables- x 1, , x n a 1,1 x 1 + a 1,2 x 2 +... + a 1,n x n = b 1 a 2,1 x 1 + a 2,2 x 2 +... + a 2,n x n = b 2 2 ... a m,1 x 1 + a m,2 x 2 +... + a m,n x n = b m

1.1.2 Solution of System of Linear Equations Definition 3 (solution())). A list (S 1, S 2, S n ) of numbers is called a solution of (2) iff (i.e. if and only if) all the equations in (2) are satisfied by substituting S 1, S 2, S n for X 1, X 2, X n. The set of all solutions of (2) is called the solution set () of (2). Two systems of linear equations are said to be equivalent () 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 3. Infinitely many solutions. Definition 4 (consistence ( )). 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. consistent inconsistent

1.1.2 Solution of System of Linear Equations Fig(a). Exactly one solution Fig(b). no solution Fig(c). Infinitely many solutions

1.1.3 Matrix Notation P.4 Matrix Notation Coefficient matrix augmented matrix The size of a Matrix: how many rows and columns it has.

1.1.3 Matrix Notation Definition 5 (matrix ( )). A table of numbers with m rows ( ) and n columns ( ) 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 matrixaugmented matrix

1.1.4 Solving a Linear System P.5 Basic strategy ( ). To replace one system with an equivalent system (one with the same solution set) that is easier to solve Three basic operations ( elementary operation( ) to simplify a linear system 1. replace( ) one equation by the sum of itself and a multiple of another equation 2. interchange( ) two equations 3. Scaling( ) all the terms in an equation by a nonzero constant

Solving a Linear System 4*[eq.1]+[eq.3] ()*[eq.2] 3*[eq.2]+[eq.3] 4*[eq.3]+[eq.2] -1*[eq.3]+[eq.1] Sol: Upper triangular Back subsitution

Sol: augmented matrix

Definition 6 (Row Equivalence ( )). 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 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 x 3, and substitute the value of x 3 into eq.2 could get x 2, then could determine x 1 from eq.1. So a solution exists; the system is consistent.

Eg:Determine if the following system is consistent: Sol: The equation 0x 1 +0x 2 +0x 3 =(5/2) is never true, so the system is inconsistent.

1.2 Row Reduction and Echelon Forms P.14 Basic concept: leading entry ( ) (row) echelon form ( ) echelon matrix ( ) reduced (row) echelon form ( ), reduced (row) echelon matrix( ) pivot position ( )

1.2.1 Echelon Forms P.14 *Definition 1:leading entry( ): 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.

The following matrices are in reduced echelon form:

Theorem 1 : Uniqueness of the Reduced Echelon Form (p.15) Each matrix is row equivalent to one and only one reduced echelon matrix. 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 U the reduced echelon form of A.

1.2.2 Pivot position( ) 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: Interchange row1 and row4 Adding multiples of the first rows below: Example 2: Row reduce the matrix A below to echelon form, and locate the pivot columns of A.

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( ) 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: 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)

1.2.4. Solution of Linear Systems P.20 augmented matrixAssociated system of equation solution Basic variable( ): any variable that corresponds to a pivot column in the augmented matrix of a system. free variable( ) all nonbasic variables. (5) (4)

Example 4: Find the general solution( ) 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

1.2.6 Existence and Uniqueness Questions P.23 (8)

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 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 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

Solutions of Linear Systems( )

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 ( ), linear combination ( ), Span ( )

1.3 Vector Equations P.28 Vectors in R 2 Geometric Description of R 2 Vectors in R 3 Vectors in R n 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, ) A matrix with only one column is called a column vector, or simply a vector. 1.3.1 Vectors in R 2 A two-dimensional vector is a pair of numbers, surrounded by brackets( ).

1.3 Vector Equations Vectors in R 2 Notation: Different people use different notation for vector. v (boldface), (use arrows)

Vectors in R 2

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( ) : 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 R 2 Vector as points Vectors with arrows

Parallelogram Rule ( )For Addition If u and v in are represented as points in the plane, then u+v corresponds to the fourth vertex of the parallelogram whose other vertices are u,0 and v. Fig. The parallelogram rule

1.3.2 Vectors in R 3 - vectors in R 3 are 31 column matrices with three entries. - represented geometrically by points in a 3D coordinate space Fig. Scalar multiples in R 3

1.3.3 Vectors in R n (n ) R n denotes the collection of all lists of n real numbers - written as n1 column matrices - zero vector

Algebraic Properties( ) of R n For all u, v, w in R n and all scalars c and d: 0

1.3.4. Linear Combination( ) p. 32 Definition 4(Linear combination):Given vectors v 1, v 2,,v p in R n and given scalars c 1, c 2,,c p, the vector y defined by is called a linear Combination of v 1, v 2,,v p with weight ( ) c 1, c 2,,c p. eg. ?

Example

Let and Determine whether b can be generated as a linear combination of a 1 and a 2. ( b a 1, a 2 ) That is, determine whether x 1 and x 2 exist such that x 1 a 1 + x 2 a 2 = b. Sol. and

Get the system: Solve the system: so:Hence b is a linear combination of a 1 and a 2

Facts A vector equation has the same solution set as the linear system whose augmented matrix is In particular, b can be generated by a linear combination of a 1, ,a n if and only if there exists a solution to the linear system corresponding to

1.3.5 Span ( ) P.35 b Span{ v 1, v 2,, v n }, x 1 v 1 +x 2 v 2 +..+ x n v n =b

A Geometric Description of Span{v} and Span{u,v}

Eg: A company manufactures two products. For $1.00 worth of product B, the company spend $.45 on materials, $.25 on labor, and $.15 on overhead. For $1.00 worth of product C, the company spend $.40 on materials, $.30 on labor, and $.30 on overhead. Let A.what economic interpretation can be given to the vector 100b? B. Suppose the company wishes to manufacture x1 dollars worth of product B and x2 dollars worth of product C. Give a vector that describes the various costs the company will have.

So l. A. The vector 100b list the various costs for producing $100 worth of product B, $45 for material, $25 for labor, and $15 for overhead. B. The costs of manufacturing x 1 dollars worth of B are given by the vector x 1 b, and the costs of manufacturing x 2 dollars worth of C are given by the vector x 2 c. Hence the total costs for both products by the vector x 1 b+x 2 c

CHAPTER 1 Linear Equations in Linear Algebra. §1.1 Systems of Linear Equations Basic concept linear equation( 线性方程 ) system of linear equations( 线性方程.

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