Linear Algebra

线性代数

Linear Algebra

• Chapter 1 Linear Equations 线性方程（组）• Chapter 2 Matrix Algebra 矩阵代数• Chapter 3 Determinants 行列式• Chapter 4 Vector Spaces 向量空间• Chapter 5 Eigenvalues and Eigenvectors

特征值、特征向量

Linear Equations

• 1.1 System of linear Equations • 1.2 Row Reduction and Echelon Forms • 1.3 Vector Equations • 1.4 The Matrix Equation Ax = b • 1.5 Solution Sets of Linear Systems • 1.7 Linear Independence

System of linear Equations

• What is a linear equation? 线性方程• A:

• Coefficients ： a1,…, an

• Exp:

1 1 2 2 n na x a x a x b

1 2 43 4 6 9x x x

• What is a system of linear equations ( linear system ) ? 线性方程组，线性系统

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x ba x a x a x b

a x a x a x b

• Solution 解• Solution set 解集• Equivalent 等价• Exp:

1 2

1 2

3 2 65 4 8x xx x

1 2

1 2

2 72 4 14x xx x

• A system of linear equations has either – 1. no solution, or – 2. exactly one solution, or – 3. infinitely many solutions.

1 2 1 2 1 2

1 2 1 2 1 2

2 1, 2 1, 2 1,2 3. 3 3. 2 1.

x x x x x xx x x x x x

• Consistent ( 相容的，有解的 ) inconsistent ( 不相容的，无解的 )

• A linear system is consistent if it has at least one solution.

Matrix Notion

• A rectangular array --- Matrix 矩阵 11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

m m mn n m

a x a x a x ba x a x a x b

a x a x a x b

• Coefficient matrix ( matrix of coefficients ) 系数矩阵11 12 1

21 22 2

1 2

n

n

m m mn

a a aa a a

a a a

• Augmented matrix 增广矩阵11 12 1 1

21 22 2 2

1 2

n

n

m m mn m

a a a ba a a b

a a a b

• Size ( of a matrix ) m by n 一个矩阵是 m 乘 n 的

Elementary Row Operations行初等变换 • ( Replacement ) Replace one row by the

sum of itself and a multiple of another row.

• ( Interchange ) Interchange two rows.• ( Scaling ) Multiply all entries in a row by

a nonzero constant.

• 替换 交换 倍乘

1 2 1 00 2 8 84 5 9 9

1 2 1 00 2 8 84 5 9 9

1 2 1 00 1 4 40 0 1 3

Exp1: Determine if the linear system is consistentwhich the augmented matrix is

..….

1 2 1 00 2 8 80 3 13 9

1 2 1 00 1 4 40 3 13 9

Solution:

0 1 4 82 3 2 15 8 7 1

0 1 4 82 3 2 15 8 7 1

2 3 2 10 1 4 80 0 0 2.5

Exp2: Determine if the linear system is consistentwhich the augmented matrix is .

… …

2 3 2 10 1 4 85 8 7 1

2 3 2 10 1 4 80 0.5 2 1 5

.

Solving a Linear System Exp3:

1 2 3

2 3

1 2 3

2 02 8 8

4 5 9 9

x x xx x

x x x

1 0 0 290 1 0 160 0 1 3

1 2 1 00 2 8 80 3 13 9

1 2 3

2 3

2 3

2 02 8 83 13 9

x x xx xx x

1

2

3

2916

3

xx

x

1 2 1 00 2 8 84 5 9 9

1 2 1 00 1 4 40 3 13 9

1 2 1 00 1 4 40 0 1 3

1 2 0 30 1 0 160 0 1 3

1 2 3

2 3

3

2 04 4

3

x x xx x

x

27403212

321

321

321

xxxxxxxxx

Ex4 ： Find the general solutions of the linear system

201

174132121

A

22

1

310310121

02

1

000310121

023

000310501

023

000310

501

.23

35

3

32

31

freeisxxxxx

Ex5 ： Find the general solutions of the linear system

2132

130

4321

4321

4321

xxxx

xxxxxxxx

2/110

32113111

1111A

2/110

21004200

1111

02/1

0

00002100

1111

02/12/1

000021001011

.,212

21

42

43

421

freearexx

xx

xxx