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# 大學線性代數初步

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. 學 大 大 ( ) 線性

, 學 , 線性代數

. , 線性代數

, .

, . 學

. , (Question).

, 大

. , 線性代數 . ,

, ,

, . , 性

, . , .

v

(3)

## Systems of LinearEquations

. 大 (

) . ,

. ,

, 性 . 性 ,

.

2.1.

n n 數 (variable) (linear equation).

2x1+ 5x2− x3+ x4= 1 4 ( 5

). n

a1x2+··· + anxn= b,

a1, . . . , an b 數, xi 數. n

, (system of linear equations).

a11x1 + a12x2 + ··· + a1nxn = b1

a21x1 + a22x2 + ··· + a2nxn = b2 ...

am1x1 + am2x2 + ··· + amnxn = bm

m n . a11x1+ a12x2+··· + a1nxn = b1 , a21x1+ a22x2+··· + a2nxn= b2 , 1≤ i ≤ m , i ai1x1+ ai2x2+··· + ainxn= bi, ( m )

am1x1+ am2x2+··· + amnxn= bm. ai j, bi 數, 數

21

(4)

, ,

A =





a11 a12 ··· a1n

a21 a22 ··· a2n

... ... ... ... am1 am2 ··· amn



, x =



 x1 x2 ... xn



, b =



 b1 b2 ... bm



,

Ax = b . A ai j A entry.

A entry 數 數, A

column ( ). row 數 , row

row, row row, . column 數 ,

column column, column column,

. 大 A row , row

, row , . column

, A m row n column, m× n matrix.

x x, b column vector ( )

. 前大 .

3x1− 2x2+ 9x4 = 4

2x1+ 2x2− 4x4 = 6 (2.1)

[ 3 −2 0 9

2 2 0 −4

]



x1

x2

x3 x4



 = [ 4

6 ]

[ 0 0

]

column x3 數 0.

? 線性代數

. 1.3 span . u = (1,−1,2,2),v = (3,1,−1,2)

w = (1, 0, 1, 0) Span(u, v) c1, c2∈ R w = c1u + c2v.

,

(1, 0, 1, 0) = c1(1,−1,2,2) + c2(3, 1,−1,2).

x1 + 3 x2 = 1

−x1 + x2 = 0 2 x1 x2 = 1 2 x1 + 2 x2 = 0

(5)

2.1. 23

. x = (x1, x2, x3, x4) u· x = 0 v· x = 0, x1 − x2 + 2 x3 + 2 x4 = 0

3 x1 + x2 x3 + 2 x4 = 0.

, .

, :

(1)

(2) 數

(3) 數

: 步 ?

? 步 ,

.

a11x1 + a12x2 + ··· + a1nxn = b1

a21x1 + a22x2 + ··· + a2nxn = b2

...

am1x1 + am2x2 + ··· + amnxn = bm

, augmented matrix ( )





a11 a12 ··· a1n b1

a21 a22 ··· a2n b2 ... ... . .. ... ... am1 am2 ··· amn bm





(2.1) augmented matrix

[ 3 −2 0 9 4

2 2 0 −4 6

]

augmented matrix [A| b] Ax = b.

:

(1) row

(2) row 數

(3) row 數 row.

(6)

matrix 前 步 .

augmented matrix [A| b]A elementary row operation echelon form.

echelon form. row

0 row leading entry. 數 entry

variable 數, leading entry variable xi 數,

i column.

 1 2 1 1 4

0 0 5 0 2

0 0 1 −1 1

row leading entry 1 row 1,

entry 5 1 x3.

echelon form leading entry row ( row

. 言 , row leading entry xi, row

lading entry xj, i < j. echelon form, 3 row

2 row leading entry x3, .

 1 2 −1 0

0 0 0 0

0 0 3 0

,

 0 1 1 2 0 0 2 −1 3 0 0 0

echelon form, 前 0 row , 3



0 2 1 1 4

0 0 3 0 2

0 0 0 −1 1

0 0 0 0 0



echelon form. echelon form , row leading entry

pivot, pivot pivot variable.

augmented matrix [A| b] elementary row operation [A| b]

A echelon form . A . A row 0; A

row 0. .

(1) A row 0: consistent, .

.

(7)

2.1. 25

(a) 數 (variable) xi pivot variable. pivot 數

 2 1 1 4

0 3 1 2

0 0 −1 1

echelon form pivot variable x1, x2, x3

“代 ” . 前 augmented matrix

2x1 +x2 +x3 = 4 3x2 +x3 = 2

−x3 = 1

−x3= 1 x3=−1. x3=−1 代

3x2+x3= 2, 3x2−1 = 2, x2= 1. x3=−1,x2= 1 代 2x1+x2+x3= 4, x1= 2. x1= 2, x2= 1, x3=−1.

(b) variable xi pivot variable. 數

 2 1 3 1 4

0 3 3 1 2

0 0 0 −1 1

echelon form pivot variable x1, x2, x4

x1, x2, x3, x4. .

, free variables. free variable

pivot variable variable. 前 , x3 free variable. Free

variable , free variables

. augmented matrix

2x1 +x2 +3x3 +x4 = 4 3x2 +3x3 +x4 = 2

−x4 = 1

free variable x3 數 t ( 數 t ∈ R).

−x4= 1 x4=−1. x3= t, x4=−1 代

3x2+ 3x3+ x4= 2, 3x2+ 3t− 1 = 2, x2= 1− t. x2= 1− t,x3= t, x4=−1 代 2x1+ x2+ 3x3+ x4= 4, x1= 2−t. x1= 2−t,x2=

1−t,x3= t, x4=−1, t 數. t 數,

.

(2) A row 0: , :

(8)

(a) A row 0 b row 0.

[A| b] =

 2 1 1 4 0 3 1 2 0 0 0 1

A row 0, b row 1.

inconsistent, . augmented matrix row

0x1+ 0x2+ 0x3= 1

x1, x2, x3 代 數 0x1+ 0x2+ 0x3= 1, .

(b) A 0 row, b row 0.

 2 1 4 0 3 2 0 0 0

,

 2 1 3 1 4 0 3 3 1 2 0 0 0 0 0

augmented matrices .

consistent. 0 row, 前

augmented matrices [ 2 1 4

0 3 2 ]

,

[ 2 1 3 1 4 0 3 3 1 2

]

. 前 (1) A row 0

.

, pivot 數 variables ( 數) 數 .

A echelon form , column pivot (

leading term ), pivot 數 column 數.

A column 數 variables 數, pivot 數

variables 數. row pivot, pivot 數

Question 2.1. n variables m .

(1)(a) m = n ; (1)(b) m < n

;

Ax = b 步 . augmented matrix

[A| b] elementary row operations [A| b] A echelon form .

. elementary row

operations A echelon form A, .

(9)

2.2. Elementary Row Operations 27

2.2. Elementary Row Operations

Ax = b, augmented matrix [A| b]

elementary row operations [A| b] A echelon form .

elementary row operations echelon

form elementary row operations .

. row 數 數學 . row

echelon form, row elementary

row operations echelon form. row 3 row

elementary row operations echelon form,

4, 5, 6, . . . row . 數 row

( 10 row), ( 數 row). 數學

. k row elementary row

operations echelon form k + 1 row

elementary row operations echelon form, row

elementary row operations echelon form row

elementary row operations echelon form, 3 row ,

4 row , elementary

row operations echelon form.

, . 3 row

 0 0 1 1 1

0 1 2 1 3

0 2 2 0 −1

echelon form row leading entry ,

, row elementary row operation

 0 1 2 1 3

0 0 1 1 1

0 2 2 0 −1

row leading entry row 數 .

row −2 row elementary row operation

 0 1 2 1 3

0 0 1 1 1

0 0 −2 −2 −7

entry . row row .

(10)

row [

0 0 1 1 1

0 0 −2 −2 −7 ]

row echelon , .

, row 2 row

[ 0 0 1 1 1 0 0 0 0 −5

]

echelon form. row

 0 1 2 1 3

0 0 1 1 1

0 0 −2 −2 −7

row 2 row

 0 1 2 1 3

0 0 1 1 1

0 0 0 0 −5

echelon form.

. row .

row echelon form. row .

echelon form row leading entry ( ) row

row ( row leading entry row) row

row operation row. row leading entry

row , row leading entry , row

echelon form. row leading entry row ,

entry, row . row leading entry 0,

3 row , 數學 ,

k row , k row elementary row

operation echelon form. k + 1 row . 前 ,

leading entry row row row operation

b, row −b/a row . row leading entry

. 步 , row row leading entry

row leading entry . , row row

(11)

2.2. Elementary Row Operations 29

row, k row , 前 k row

elementary row operations echelon form, elementary row operations

row echelon form. row leading entry

row leading entry , echelon form.

elementary row operations echelon form. 大

echelon form row 0 數 elementary

row operation. echelon form row operation,

“reduced” echelon form , .

elementary row operations echelon form,

elementary row operation .

augmented matrix elementary row operations augmented matrix

. Ax = b augmented matrix

[A| b] elementary row operations [A | b],

Ax = b Ax = b . 數 A elementary

row operations A Ax = b Ax = b .

Ax = b augmented matrix [A| b] elementary row operation [A| b]

Ax = b. Ax = b (

0 數 數 )

Ax = b, Ax = b Ax = b. Ax = b

Ax = b . ,

Ax = b Ax = b. Ax = b

Ax = b . Ax = b Ax = b .

[A| b] elementary row operation [A| b],

. [A| b] elementary row operation

[A| b], .

Example 2.2.1. Solve the linear system

x2 −3x3 = −5 2x1 +3x2 −1x3 = 7 4x1 +5x2 −2x3 = 10.

augmented matrix

 0 1 −3 −5

2 3 −1 7

4 5 −2 10

.

(12)

, row, , row

 2 3 −1 7 0 1 −3 −5 4 5 −2 10

.

row leading entry x1 , row −2

row 

 2 3 −1 7

0 1 −3 −5

0 −1 0 −4

.

row 

 2 3 −1 7 0 1 −3 −5 0 0 −3 −9

.

echelon form. 數 0 row, linear system consistent.

pivot 數 variable 數, linear system . ,

row −3x3=−9, x3= 3. 代 row x2− 3x3=−5, x2= 4.

Example 2.2.2. Solve the linear system

x1 −1x2 +2x3 +3x4 = 2 2x1 +1x2 +1x3 = 1 x1 +2x2 −1x3 −3x4 = 7.

augmented matrix

 1 −1 2 3 2

2 1 1 0 1

1 2 −1 −3 7

.

, row leading entry . row −2,−1 , row

 1 −1 2 3 2

0 3 −3 −6 −3

0 3 −3 −6 5

.

row leading entry , row −1 row

 1 −1 2 3 2

0 3 −3 −6 −3

0 0 0 0 8

.

echelon form. row 0x1+0x2+0x3= 8, linear system inconsistent.

(13)

2.2. Elementary Row Operations 31

Example 2.2.3. Solve the linear system

x1 −2x2 +1x3 −1x4 = 4 2x1 −3x2 +4x3 −3x4 = −1 3x1 −5x2 +5x3 −4x4 = 3

−x1 +1x2 −3x3 +2x4 = 5.

augmented matrix



1 −2 1 −1 4

2 −3 4 −3 −1

3 −5 5 −4 3

−1 1 −3 2 5



.

, , row leading entry . row −2,−3,1 ,

, row 



1 −2 1 −1 4

0 1 2 −1 −9

0 1 2 −1 −9

0 −1 −2 1 9



.

, row leading entry , row −1,1 ,

row 



1 −2 1 −1 4

0 1 2 −1 −9

0 0 0 0 0

0 0 0 0 0



.

echelon form. 數 0 , row 0, linear system

consistent.

linear system pivot variables x1, x2, free variables x3, x4. x4= r, x3= s, 代 row x2+ 2x3− x4=−9, x2=−9 + r − 2s. 代 row x1− 2x2+ x3− x4= 4, x1=−14 + 3r − 5s. linear system

(x1, x2, x3, x4) = (−14 + 3r − 5s,−9 + r − 2s,s,r),r,s ∈ R.

row vector r, s .



x1 x2 x3

x4



 =



−14−9 0 0



 + r



 3 1 0 1



 + s



−5−2 1 0



,r,s ∈ R.

linear system 步 echelon form reduced echelon

form. Reduced echelon form echelon form, .

pivot 1. pivot 0. , echelon

form pivot 0 reduced echelon form pivot column,

1 0.

A =

 1 2 0 0 0 0 3 6 0 0 0 0

, B =

 1 1 3 0 0 1 1 2 0 0 1 −1

(14)

reduced echelon form A=

 1 2 0 0 0 0 1 2 0 0 0 0

, B=

 1 0 0 0 0 1 0 3 0 0 1 −1

reduced echelon form. echelon form elementary row operations

reduced echelon form. , row pivot a ( a̸= 0),

row 1/a, row pivot 1 . A echelon form

row 1/3, A reduced echelon form. pivot 1 ,

row 數 row pivot column

0. B echelon form row −3, −1

row row,

 1 1 0 3 0 1 0 3 0 0 1 −1

. row −1 row,

B reduced echelon form. echelon form,

echelon form echelon form reduced echelon form . elementary row operations reduced echelon form,

. reduced echelon form

row row pivot , free variables ( pivot variable entry

0), . Bx = O

x1 = 0

x2 +3x4 = 0 x3 −x4 = 0

x4 free variable, x4= t, 代 row x3= t. 代 row x2=−3t.

row x1= 0. (x1, x2, x3, x4) = (0,−3t,t,t) = t(0,−3,1,1),t ∈ R.

reduced echelon form ,

reduced echelon form echelon form 步 , echelon form

. echelon form Gauss method, reduced

echelon form Gauss-Jordan method.

Example 2.2.4. Example 2.2.1 linear system, echelon form

 2 3 −1 7 0 1 −3 −5 0 0 −3 −9

.

row −1/3

 2 3 −1 7 0 1 −3 −5

0 0 1 3

.

row 3, 1 , row

 2 3 0 10 0 1 0 4 0 0 1 3

.

(15)

2.3. The Rank of a Matrix 33

row −3 row

 2 0 0 −2

0 1 0 4

0 0 1 3

.

row 1/2 reduced echelon form

 1 0 0 −1

0 1 0 4

0 0 1 3

,

(x1, x2, x3) = (−1,4,3).

Example 2.2.3 linear system, echelon form



1 −2 1 −1 4

0 1 2 −1 −9

0 0 0 0 0

0 0 0 0 0



.

row 2 row reduced echelon form



1 0 5 −3 −14

0 1 2 −1 −9

0 0 0 0 0

0 0 0 0 0



.

x4, x3 free variables, x4= r, x3= s, 代 row x2=−9 + r − 2s. 代 row x1=−14 + 3r − 5s.

2.3. The Rank of a Matrix

elementary row operations echelon form , 0 row

.

echelon form , echelon form ,

echelon form , pivot 數 .

. , .

Deﬁnition 2.3.1. A . A elementary row operations echelon

form pivot 數 r, r A rank. rank(A) = r .

Ax = b, A m× n matrix rank(A) = r. r = m,

A elementary row operations echelon form row 0, 2.1

(1) . b , Ax = b consistent, .

r < m ( r > m ) ? A elementary row operations

echelon form A , row 0 . b [A| b]

elementary row operations [A| b] , b echelon form A 0

(16)

row 0 ( 2.1 (2)(a) ), Ax = b inconsistent,

Ax = b . b b echelon form A 0 row

0, elementary row operations b

Ax = b . r < m b Ax = b .

b Ax = b , Ax = b .

m× n matrix A, b Ax = b , Ax = b ?

b =

 b1

... bm

 Rm , [A| b] .

elementary row operations A echelon form A, [A| b] [A| b].

elementary row operations, b=

 b1

... bm

 bi

b1, . . . , bm 數 . rank(A) = r, echelon form A 前 r row

0, A m− r row 0. Ax = b , 2.1

(2) , b m− r br+1, . . . , bm 0. Ax = b

, b =

 b1

... bm

 br+1, . . . , bmm− r b1, . . . , bm

0. 言 , Ax = b b m− r

b1, . . . , bmm . .

Proposition 2.3.2. m×n matrix A rank(A) = r. A.

(1) r = m b∈ Rm, Ax = b .

(2) r < m, (m− r) × m matrix B, Ax = b y = b

By = O .

Proposition 2.3.2 (2) By = O y = b

Ax = b , Ax = b , y = b By = O .

By = O Aconstrain equations.

A constrain equations ,

constrain equations, , Ax = b consistent b

.

Example 2.3.3. A =



1 −1 1

3 2 −1

1 4 −3

3 −3 3



A

constrain equations. elementary row operation, rank(A) = 2, A

constrain equations (4− 2) × 2 .

(17)

2.3. The Rank of a Matrix 35

augmented matrix



1 −1 1 b1 3 2 −1 b2

1 4 −3 b3

3 −3 3 b4



. 1-st row −3, −1, −3

2-nd, 3-rd, 4-th rows



1 −1 1 b1

0 5 −4 b2− 3b1

0 5 −4 b3− b1

0 0 0 b4− 3b1



. 2-nd row −1 3-rd

row



1 −1 1 b1

0 5 −4 b2− 3b1

0 0 0 b3− b2+ 2b1 0 0 0 b4− 3b1



. b =



b1

b2

b3 b4



Ax = b consistent b1, b2, b3, b4 constrain equations

2b1 −b2 +b3 = 0

−3b1 +b4 = 0 .

B =

[ 2 −1 1 0

−3 0 0 1

]

, 2× 4 matrix By = O A

By = O



y1 y2

y3

y4



 = r



 0 1 1 0



 + s



 1 2 0 3



, r,s ∈ R.

Ax = b consistent b

Span(



 0 1 1 0



,



 1 2 0 3



).

Question 2.2. A m× n matrix. n < m , b∈ Rm

Ax = b inconsistent. 數 數 數 , b

Ax = b .

rank 數 . A m× n matrix rank(A) = r.

r = n , x1. . . . , xn n variable pivot variable. Ax = b

( 2.1 (2)(a) ); Ax = b , free variable, x1, . . . , xn

, Ax = b . r < n, n− r free variables,

Ax = b consistent , free variables Ax = b ,

. .

Proposition 2.3.4. m× n matrix A. rank(A) = r Ax = b consistent.

(1) r = n, Ax = b .

(2) r < n, Ax = b .

(18)

Question 2.3. A m× n matrix. n > m , b∈ Rm

Ax = b . 數 數 數 , Ax = b

; .

consistent, Ax = O linear system.

x1= 0, . . . , xn= 0 . Ax = O linear system

homogeneous system. .

Corollary 2.3.5. m× n matrix A. Ax = O x1=··· = xn= 0 rank(A) = n.

homogeneous system Ax = O x1= 0, . . . , xn= 0 ,

Ax = O trivial solution. x1= 0, . . . , xn= 0 Ax = O ,

Ax = O nontrivial solution. Ax = O nontrivial

solution; nontrivial solution.

Question 2.4. A m× n matrix. Ax = O nontrivial solution rank(A)̸= n.

Ax = b consistent , Ax = b Ax = O . ,

x1= c1, . . ., xn= cn x1= c1, . . ., xn= cn Ax = b .









a11c1 + ··· + a1ncn = b1 a21c1 + ··· + a2ncn = b2

...

am1c1 + ··· + amncn = bm









a11c1 + ··· + a1ncn = b1 a21c1 + ··· + a2ncn = b2

...

am1c1 + ··· + amncn = bm

a11(c1− c1) + ··· + a1n(cn− cn) = 0 a21(c1− c1) + ··· + a2n(cn− cn) = 0

...

am1(c1− c1) + ··· + amn(cn− cn) = 0 .

x1= c1− c1, . . . , xn = cn− cn Ax = O . , x1= c1, . . ., xn= cn Ax = b x1 = u1, . . ., xn= un Ax = O , x1 = c1+ u1, . . ., xn= cn+ un Ax = b . .

Proposition 2.3.6. Ax = b consistent x = c . x = c

Ax = b x = c− c Ax = O .

Proposition 2.3.6, .

Corollary 2.3.7. A m× n matrix. .

(1) Ax = b consistent Ax = b .

(2) Ax = O nontrivial solution.

(19)

2.4. 37

(3) rank(A) = n.

Proof. ( ) , .

(1)⇒ (2), (2) ⇒ (3) (3)⇒ (1) . , (2)⇒ (1)

(2)⇒ (3) (3)⇒ (1) . statement

, 數學 .

(1)⇒ (2): , x1= u1, . . ., xn= un Ax = O nontrivial solution x1= c1, . . ., xn= cn Ax = b , Proposition 2.3.6 x1= c1+ u1, . . ., xn= cn+ un

Ax = b . Ax = b , Ax = O nontrivial solution.

(2)⇒ (3): Ax = O nontrivial solution, Ax = O , Corollary 2.3.5 rank(A) = n.

(3)⇒ (1): Proposition 2.3.4 (1). 

A n× n .

Corollary 2.3.8. A n× n square matrix. .

(1) b∈ Rn, Ax = b .

(2) b∈ Rn, Ax = b .

(3) Ax = O nontrivial solution.

(4) rank(A) = n.

Proof. , .

. .

(1)⇒ (4), (4) ⇔ (3), (4) ⇒ (2) (2)⇒ (1).

(1)⇒ (4): Proposition 2.3.2 (1), b∈ Rn, Ax = b rank(A) A row 數, rank(A) = n.

(4)⇔ (3): Corollary 2.3.7 (3)⇔ (2) .

(4)⇒ (2): rank(A) = n rank(A) A row 數, Proposition 2.3.2 (1) b∈ Rn, Ax = b . Corollary 2.3.7 (3)⇒ (1) .

(2)⇒ (1): (2) (1) , . 

Corollary 2.3.8 n× n matrix ( rank n) nonsingular matrix, rank n n× n matrix singular matrix.

2.4.

matrix 數 echelon form , ,

echelon form . echelon form

(20)

pivot . ,

, .

, ,

.

column vector

Proposition 9.4.2, A orthogonal diagonalizable, Spectral Theorem.. Theorem 9.4.6

Theorem 8.2.6 (3) elementary column operation.. determinant elementary row

, A echelon form ( reduced echelon form) pivot column vectors.. elementary row operations column

minimal element; vector space linearly independent set

( diagonalizable), linear operator rational form Jordan form. cyclic subspace

elementary row operations reduced echelon form,.. echelon form Gauss