大學線性代數初步
大學數學
數學 大 學 線性代數 , 大
數 .
大 學 線性代數學 ,
學 線性代數 . 大學線性代數
. 學 大 大 ( ) 線性
代數 , . 大學 線性代數 , 大
學 數學 . 大 大 數學
, 學 , 線性代數
. , 線性代數
數學 . , .
, .
學 , 數學 學 , 線性
代數 . 線性代數 .
, . 學
. , (Question).
, 大
. , 線性代數 . ,
學 線性代數 學 線性代數 , .
, ,
代. , .
, . , 性
, . , .
v
Chapter 2
Systems of Linear Equations
. 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
, ,
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
數 . row ( ),
column ( ). row 數 , row
row, row row, . column 數 ,
column column, column column,
. 大 A row , row
, row , . column
數, column 數 x1 數, column 數 x2
數, . m n 數
, 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 ]
數
[ 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
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.
, .
學 ,
, :
(1)
(2) 數
(3) 數
數 , .
: 步 ?
? 步 ,
大 , .
.
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
]
言 , Ax = b , [A| b] matrix.
augmented matrix [A| b] Ax = b.
步 , elementary row operation ( ) augmented matrix. elementary row operation
:
(1) row
(2) row 數
(3) row 數 row.
大 augmented matrix augmented
matrix 前 步 .
augmented matrix [A| b] 數 A elementary row operation echelon form.
echelon form. row
0 row leading entry. 數 entry
variable 數, leading entry variable xi 數,
leading entry xi . , leading entry
i column.
1 2 1 1 4
0 0 5 0 2
0 0 1 −1 1
row leading entry 1 row 1,
row leading entry x1 , row row leading
entry 5 1 x3.
echelon form leading entry row ( row
0) , leading entry row leading entry
. 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
row leading entry 2 row leading entry .
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, .
.
2.1. 25
(a) 數 (variable) xi pivot variable. pivot 數
數 數 ( 數 A column 數).
2 1 1 4
0 3 1 2
0 0 −1 1
echelon form pivot variable x1, x2, x3
數 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
數 pivot 數.
2 1 3 1 4
0 3 3 1 2
0 0 0 −1 1
echelon form pivot variable x1, x2, x4 數
x1, x2, x3, x4. .
, free variables. free variable
pivot variable variable. 前 , x3 free variable. Free
variable , free variables
數, 代
. 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: , :
(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 .
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
數 ( 數 row 數).
Question 2.1. n variables m . 前
(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′, .
2.2. Elementary Row Operations 27
2.2. Elementary Row Operations
前 Ax = b, augmented matrix [A| b]
elementary row operations [A′| b′] A′ echelon form .
elementary row operations echelon
form elementary row operations .
數學 echelon form.
數學 , ,
. 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
row row leading entry row leading
entry . row row .
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
數學 ( row 數 ). ,
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
leading entry . row leading entry
row ( row leading entry row) row
row operation row. row leading entry
row leading entry . row leading entry
row , row leading entry , row
leading entry row leading entry ,
echelon form. row leading entry row ,
row −b/a, a row leading entry b row leading
entry, row . row leading entry 0,
leading entry , echelon form.
3 row , 數學 ,
k row , k row elementary row
operation echelon form. k + 1 row . 前 ,
leading entry row row row operation
row. row leading entry a. leading entry
row leading entry row , row leading entry
b, row −b/a row . row leading entry
. 步 , row row leading entry
row leading entry . , row row
2.2. Elementary Row Operations 29
leading entry row leading entry .
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′],
A′x = b′ Ax = b . 數 A elementary
row operations A′ A′x = b Ax = b .
Ax = b augmented matrix [A| b] elementary row operation [A′| b′]
A′x = b′. Ax = b (
0 數 數 )
A′x = b′, Ax = b A′x = b′. Ax = b
A′x = b′ . ,
A′x = b′ Ax = b. A′x = b′
Ax = b . Ax = b A′x = 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
.
, row leading entry . row leading entry ,
, 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
.
數 echelon form, row x2 entry . row
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.
代 row 2x1+ 3x2− x3 = 7, x1 =−1. linear system (x1, x2, x3) = (−1,4,3).
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.
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
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), . B′x = 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
.
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
數 ( pivot 數) , rank.
數 rank .
.
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
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′=
b′1
... b′m
b′i
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 b′r+1, . . . , b′m 0. Ax = b
, b =
b1
... bm
b′r+1, . . . , b′m 代 m− r b1, . . . , bm 數
0. 言 , Ax = b b m− r
b1, . . . , bm 數 m . .
Proposition 2.3.2. m×n matrix A rank(A) = r. A 數 .
(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 A 數 constrain 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 數
constrain equations. elementary row operation, rank(A) = 2, A
constrain equations (4− 2) × 2 .
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
數 constrain equations.
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 .
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= c′1, . . ., xn= c′n Ax = b .
a11c1 + ··· + a1ncn = b1 a21c1 + ··· + a2ncn = b2
...
am1c1 + ··· + amncn = bm
a11c′1 + ··· + a1nc′n = b1 a21c′1 + ··· + a2nc′n = b2
...
am1c′1 + ··· + amnc′n = bm
a11(c1− c′1) + ··· + a1n(cn− c′n) = 0 a21(c1− c′1) + ··· + a2n(cn− c′n) = 0
...
am1(c1− c′1) + ··· + amn(cn− c′n) = 0 .
x1= c1− c′1, . . . , xn = cn− c′n 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.
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.
學 . elementary row operations augmented
matrix 數 echelon form , ,
echelon form . echelon form
pivot . ,
, .
, ,
.
