线性代数 英文讲义

Chapter 1 Matrices and Systems of Equations

Linear systems arise in applications to such areas as engineering, physics, electronics, business, economics, sociology（社会学）, ecology （生态学）, demography(人口统计学), and genetics（遗传学）, etc. §1. Systems of Linear Equations

New words and phrases in this section:

Linear equation 线性方程

Linear system，System of linear equations 线性方程组

Unknown 未知量

Consistent 相容的

Consistence 相容性

Inconsistent不相容的

Inconsistence 不相容性

Solution 解

Solution set 解集

Equivalent 等价的

Equivalence 等价性

Equivalent system 等价方程组

Strict triangular system 严格上三角方程组

Strict triangular form 严格上三角形式

Back Substitution 回代法

Matrix 矩阵

Coefficient matrix 系数矩阵

Augmented matrix 增广矩阵

Pivot element 主元

Pivotal row 主行

Echelon form 阶梯形

1.1 Definitions

A linear equation (线性方程) in n unknowns（未知量）is

1122...

n n

a x a x a x b

+++=

A linear system of m equations in n unknowns is

1111221121122222

11

22...... ......

...n n n n m m m n n m a x a x a x b a x a x a x b a x a x a x b

+++=⎧⎪

+++=⎪⎨

⎪⎪+++=⎩ This is called a m x n (read as m by n) system.

A solution to an m x n system is an ordered n-tuple of numbers （n 元数组）12(,,...,)n x x x that satisfies all the equations.

A system is said to be inconsistent （不相容的） if the system has no solutions.

A system is said to be consistent （相容的）if the system has at least one solution.

The set of all solutions to a linear system is called the solution set

（解集）of the linear system.

1.2 Geometric Interpretations of 2x2 Systems

1111221

2112222

a x a x

b a x a x b +=⎧⎨

+=⎩ Each equation can be represented graphically as a line in the plane. The ordered pair 12(,)

x x will be a solution if and only if it lies on both

lines.

In the plane, the possible relative positions are

(1) two lines intersect at exactly a point; (The solution set has exactly one element)

(2)two lines are parallel; (The solution set is empty)

(3)two lines coincide. (The solution set has infinitely many

elements)

The situation is the same for mxn systems. An mxn system may not be consistent. If it is consistent, it must either have exactly one solution or infinitely many solutions. These are only possibilities.

Of more immediate concerns is the problem of finding all solutions to a given system.

1.3 Equivalent systems

Two systems of equations involving the same variables are said to be equivalent（等价的，同解的）if they have the same solution set.

To find the solution set of a system, we usually use operations to reduce the original system to a simpler equivalent system.

It is clear that the following three operations do not change the solution set of a system.

(1)Interchange the order in which two equations of a system are

written;

(2)Multiply through one equation of a system by a nonzero real

number;

(3)Add a multiple of one equation to another equation. (subtract