1. CHAPTER 1 Linear Equations in Linear Algebra

2. §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)

3. 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 + a2x anxn = b (1)
where b and the coefficients a1,…,an are real or complex numbers.
eg.

4. 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+ a1,2x a1,nxn = b1
a2,1x1+ a2,2x a2,nxn = b （2）
. . .
am,1x1+ am,2x am,nxn = bm

6. 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.

7. 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

8. 1.1.2 Solution of System of Linear Equations
Fig(a). Exactly one solution
Fig(b). no solution Fig(c). Infinitely many solutions 8

9. 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.

10. 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

11. 1.1.4 Solving a Linear System P.5
Basic strategy (基本策略).
To replace one system with an equivalent system (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

12. Solving a Linear System ：
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

13. augmented matrix
Sol:

15. 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)

16. 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?

17. 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.

18. Eg:Determine if the following system is consistent:
Sol:
The equation 0x1+0x2+0x3=(5/2) is never true, so the system is inconsistent.

19. §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) 19

20. 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.

21. The following matrices are in echelon form(upper triangular matrix):

22. 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.

23. The following matrices are in reduced echelon form:

24. 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 U the reduced echelon form of A. 24

25. 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.

26. Example 2: Row reduce the matrix A below to echelon form, and locate the pivot columns of A.
Sol:
Interchange row1 and row4
Adding multiples of the first
rows below:

27. Note： There is no more than one in any row. There is
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.

28. 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

29. Example 3: Transform the following matrix into reduced echelon:
Sol:
Step1:
Step2:
Step3:

30. 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.

31. 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.

32. Example 4: Find the general solution(通解) of the following linear system

33. The associated system now is
The general solution is:
(7)

34. 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. 34

35. 1.2.5 Parametric Descriptions of Solution Sets P.22
Back-Substitution
A computer program would solve system by back-substitution 35

36. 1.2.6 Existence and Uniqueness Questions P.23
(8)

38. 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

39. Solutions of Linear Systems(线性方程组的解)

42. 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.

43. linear combination (线性组合), Span (张)
§1.3 Vector Equations P.28
Basic concept:
column vector (列向量),
linear combination (线性组合),
Span (张)

44. §1.3 Vector Equations P.28 Vectors in R2 Geometric Description of R2
Vectors in Rn
Linear Combination
A Geometric Description of Span{v} and
Span{u,v}
Linear Combinations in Applications

45. 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).

46. §1.3 Vector Equations Vectors in R2 Notation:
Different people use different notation for vector.
v (boldface), (use arrows)

47. Vectors in R2

48. 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 = ≠

49. Geometric Description of R2
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

51. Parallelogram Rule (平行四边形)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

52. - vectors in R3 are 3×1 column matrices with three entries.
- represented geometrically by points in a 3D coordinate space
Fig. Scalar multiples in R3

53. 1.3.3 Vectors in Rn (n 维向量空间)
Rn denotes the collection of all lists of n real numbers
- written as n×1 column matrices
- zero vector

54. Algebraic Properties(代数性质) of Rn
For all u, v, w in Rn and all scalars c and d:
0向量存在性
负向量存在性
数乘与加法的协调

55. 1.3.4. Linear Combination(线性组合) p. 32
Definition 4(Linear combination):Given vectors v1, v2,…,vp in Rn and given scalars c1, c2,…,cp, the vector y defined by
is called a linear Combination of v1, v2,…,vp with weight (权重)c1, c2,…,cp.
eg.
零向量是任一向量组的一个线性组合?

57. Example

58. Let and Determine whether b can be generated as a linear
combination of a1 and a2.(确定b 是能表示为a1 , a2的线性组合？) That is, determine whether x1 and x2 exist such that x1a1 + x2a2 = b.
Sol.
and

59. Hence b is a linear combination of a1 and a2
Get the system:
Solve the system:
so:
Hence b is a linear combination of a1 and a2

60. 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 a1,…,an if and only if there exists a solution to the linear system corresponding to

61. 向量方程_结论

62. x1v1+x2v2+…..+ xn vn =b 1.3.5 Span (生成或张成) P.35
要判断向量b 是否属于Span{ v1, v2,…, vn },即判断向量方程
x1v1+x2v2+…..+ xn vn =b
是否有解？

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

68. Eg: A company manufactures two products. For $1
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
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.

69. Sol. The vector 100b list the various costs for producing $100
worth of product B, $45 for material, $25 for labor, and
$15 for overhead.
The costs of manufacturing x1 dollars worth of B are given by the vector x1b, and the costs of manufacturing x2 dollars worth of C are given by the vector x2c. Hence the total costs for both products by the vector x1b+x2c B.