大學線性代數初步

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 3

Matrix

, 性

. ,

. , 步 .

3.1.

. 數 ( )

(row) ( ) (column) 數 . m row n column 數

, m× n matrix. , n× n matrix ( row 數

column 數), square matrix. , Mm×n m× n

. 大 .

A =



 1 0 2 3 0 1 5 8 2 1 1 0



, (3.1)

A 3× 4 matrix, A∈ M3×4. ,

A = [a_{i j}] . A i row j column

ai j , (i, j)-th entry. A = [ai j] m× n ,

1≤ i ≤ m 1≤ j ≤ n. (3.1) A, A = [a_{i j}], a_{1 1}= 1, a_{1 2}= 0, a_{1 3}= 2, a_{1 4}= 3,

a_{2 1}= 0, a_{2 2}= 1, a_{2 3}= 5, a_{2 4}= 8, a3 1= 2, a3 2= 1, a3 3= 1, a3 4= 0.

, A row column ,

A row vectors column vectors. A = [ai j] i row

row vector _ia , j column column vector a_j .

(3.1) A,

1a =[

1 0 2 3] , ₂a[

0 1 5 8]

, ₃a =[

2 1 1 0]

39

a₁=



 1 0 2



, a₂=



 0 1 1



, a₃=



 2 5 1



, a₄=



 3 8 0



.

, row vector column vector .

, .

( Rⁿ ).

Definition 3.1.1. A = [a_{i j}] m× n matrix A^′= [a^′_{i j}] m^′× n^′ matrix.

A = A^′ m = m^′, n = n^′ 1≤ i ≤ m 1≤ j ≤ n ai j= a^′_{i j}.

. Rⁿ

, Rⁿ 數 .

Mm×n , Mm×n

數 .

數 數 .

Mm×n , 數

. 數 數 數.

.

Definition 3.1.2. A = [ai j], B = [bi j] m× n matrix. A + B = [ci j], 1≤ i ≤ m 1≤ j ≤ n c_{i j}= a_{i j}+ b_{i j}. 數 r, rA = [d_{i j}]

1≤ i ≤ m 1≤ j ≤ n di j= rai j. Definition 3.1.2

A =







a_{1 1} a_{1 2} ··· a1 n

a2 1 a2 2 ··· a2 n

... ... . .. ... a_{m 1} a_{m 2} ··· am n





, B =







b_{1 1} b_{1 2} ··· b1 n

b2 1 b2 2 ··· b2 n

... ... . .. ... b_{m 1} b_{m 2} ··· bm n







A + B =







a_{1 1}+ b_{1 1} a_{1 2}+ b_{1 2} ··· a1 n+ b_{1 n} a_{2 1}+ b_{2 1} a_{2 2}+ b_{2 2} ··· a2 n+ b_{2 n}

... ... . .. ... a_{m 1}+ b_{m 1} a_{m 2}+ b_{m 2} ··· am n+ b_{m n}







rA =







ra_{1 1} ra_{1 2} ··· ra1 n

ra_{2 1} ra_{2 2} ··· ra2 n

... ... . .. ... ra_{m 1} ra_{m 2} ··· ram n







數 數 ,

數 數 性 . 性 ,

性 ( 數 性 ), .

3.1. 41

Proposition 3.1.3. Mm×n , 性 :

(1) A, B∈ Mm×n, A + B = B + A.

(2) A, B,C∈ Mm×n, (A + B) +C = A + (B +C).

(3) O∈ Mm×n A∈ Mm×n O + A = A.

(4) A∈ Mm×n A^′∈ Mm×n A + A^′= O.

(5) r, s∈ R A∈ Mm×n, r(sA) = (rs)A.

(6) r, s∈ R A∈ Mm×n, (r + s)A = rA + sA.

(7) r∈ R A, B∈ Mm×n r(A + B) = rA + rB.

(8) A∈ Mm×n, 1A = A.

. Chapter 1 (1, 2, 3)∈ R³

Span((1,−1,2),(2,1,−2)) , x1, x2∈ R (1, 2, 3) = x1(1,−1,2)+

x₂(2, 1,−2). column vector , x₁



 1

−1 2



 + x₂



 2 1

−2



 =



1 2 3



,

x1 +2x₂ = 1

−x1 +1x2 = 2 2x1 −2x2 = 3

. ,



 1 2

−1 1 2 −2



[ x1

x₂ ]

=



1 2 3



.



 1 2

−1 1 2 −2



 [

x1

x₂ ]



 1 2

−1 1 2 −2



[ x₁ x₂ ]

= x₁



 1

−1 2



 + x₂



 2 1

−2





, .

, .

Definition 3.1.4. A = [ai j] m× n matrix b = [b_j] n× 1 matrix ( Rⁿ column vector). a_i A i-th column

Ab =







a_{1 1} a_{1 2} ··· a1 n

a_{2 1} a_{2 2} ··· a2 n

... ... . .. ... a_{m 1} a_{m 2} ··· am n











 b₁ b₂ ... b_n





= b1a₁+ b2a₂+··· + bna_n.

, A∈ Mm×n column 數 n b∈ Mn×1 row 數 n,

Ab Ab m× 1 matrix ( R^m column vector). column

vector,

Ab = b1





 a11

a₂₁ ... a_m1





+ b₂





 a12

a₂₂ ... a_m2





+··· + bn





 a1n

a_2n ... a_mn





=







b1a11+ b2a12+··· + bna1n

b₁a₂₁+ b₂a₂₂+··· + bna_2n ...

b₁a_m1+ b₂a_m2+··· + bna_mn





. (3.2)

, a =[

a₁ a₂ ··· an

] 1×n matrix b =





 b1

b₂ ... bn





 n×1 matrix, Definition

3.1.4

a b =[

a₁ a₂ ··· an

]





 b1

b2

... bn





= b₁a₁+ b₂a₂+··· + bna_n. (3.3)

a, b Rⁿ vector, ab a, b . ,

(3.2), Ab

Ab =







1a b

2a b ...

ma b





. (3.4)

Ab m×1 matrix i-th entry ia b A i-th rowia b .

Remark 3.1.5. (3.3) a, b Rⁿ , a row vector

, b column vector , a, b , a∈ M1×n, b∈ Mn×1.

a b 1× 1 ( 數) a, b Rⁿ

. , , column vector

row vector. , column vector

row vector . , b∈ Mn×1, a∈ M1×n

b a a b Rⁿ .

, a b b a (b a n× n matrix).

, A = [ai j] m× n matrix, B = [bj k] n× l matrix. B column vector b_k∈ Mn×1, 1≤ k ≤ l,

Ab_k , AB m× l matrix, AB k-th column vector Ab_k.

大 .

A



   b₁ b₂ ··· bl



 =



   

Ab1 Ab2 ··· Abl





Ab_k m× 1 matrix, AB m× l matrix. .

3.1. 43

Definition 3.1.6. A = [a_{i j}] m× n matrix B = [b_jk] n× l matrix, AB = C = [cik] m× l matrix, 1≤ k ≤ l, C k-th column ck

c_k= Ab_k=







a1 1 a1 2 ··· a1 n

a2 1 a2 2 ··· a2 n

... ... . .. ... am 1 am 2 ··· am n











 b1 k

b_{2 k} ... bn k





= b_{1 k}a₁+ b_{2 k}a₂+··· + bn ka_n. (3.5)

, 1≤ i ≤ m, 1 ≤ k ≤ l, AB (i, k)-th entry k-th column (

Ab_k) i entry. (3.4) A i-th row _ia B k-th

column bk Rⁿ . 言 , AB = [ci k], AB (i, k)-th entry c_{i k} c_{i k}=ia b_k= ai 1b_{1 k}+ ai 2b_{2 k}+··· + ai nb_{n k}=

∑

ai jb_{j k}. (3.6)

, , column 數

row 數 .

Example 3.1.7.

A =



−2 4 3 6 2 2



,B =[

4 −1 2 5

3 0 1 1

]

AB. AB 3-rd column

Ab3=



−2 4 3 6 2 2



[ 2 1 ]

= 2a1+ 1a2= 2



−2 3 2



 + 1



4 6 2



 =



0 12

6



.

AB (2, 3) entry 12 A 2-nd row B 3-rd column R² , (3, 6)· (2,1) = 12.

AB =



−2 4 3 6 2 2



[

4 −1 2 5

3 0 1 1

]

=



4 2 0 −6

30 −3 12 21 14 −2 6 12



.

Question 3.1. a, b Rⁿ , a row vector , b column

vector , a, b , a∈ M1×n, b∈ Mn×1. ba

? a, b Rⁿ ?

大 (3.6) . (3.5) ,

初 . column ,

性 , (3.6) entry . 性

.

Proposition 3.1.8. A, A^′∈ Mm×n, B, B^′∈ Mn×l. 性 . (1) A(B + B^′) = AB + AB^′.

(2) (A + A^′)B = AB + A^′B.

Proof. A + A^′ m× n B + B^′ n× l , 數 . A = [ai j], A^′= [a^′_{i j}], B = [b_{j k}], B = [b^′_{j k}], 1≤ i ≤ m, 1≤ j ≤ n 1≤ k ≤ l.

(1) 1≤ k ≤ l , A(B + B^′) k-th column AB + AB^′ k-th column.

A(B + B^′) k-th column A B + B^′ k-th column.

, B + B^′ k-th column b_k+ b^′_k, B k-th column B^′ k-th column.

A(B + B^′) k-th column

A(bk+ b^′_k) = A







b_1k+ b^′_1k b_2k+ b^′_2k

... b_nk+ b^′_nk





= (b1k+ b^′_1k)a1+ (b2k+ b^′_2k)a2+··· + (bnk+ b^′_nk)an. (3.7)

, AB + AB^′ k-th column AB k-th column AB^′ k-th column, AB + AB^′ k-th column

A





 b1k

b2k

... bnk





+ A





 b^′_1k b^′_2k ... b^′_nk





= (b_1ka₁+ b_2ka₂+··· + bnka_n) + (b^′_1ka₁+ b^′_2ka₂+··· + b^′nka_n). (3.8)

Proposition 1.2.3 性 , (3.7) (3.8) .

(2) 1≤ k ≤ l , (A + A^′)B k-th column AB + A^′B k-th column.

(A + A^′)B k-th column A + A^′ B k-th column.

, 1≤ j ≤ n , A + A^′ j-th column a_j+ a^′_j, A j-th column A^′ j-th column. (A + A^′)B k-th column

(A + A^′)bk= (A + A^′)





 b_1k b_2k ... b_nk





= b1k(a1+ a^′₁) + b2k(a2+ a^′₂) +··· + bnk(an+ a^′_n). (3.9)

, AB + A^′B k-th column AB k-th column A^′B k-th column, AB + A^′B k-th column

A





 b_1k b2k

... b_nk





+ A^′





 b_1k b2k

... b_nk





= (b_1ka₁+ b_2ka₂+··· + bnka_n) + (b_1ka^′₁+ b_2ka^′₂+··· + bnka^′_n). (3.10)

Proposition 1.2.3 性 , (3.9) (3.10) . 

scalar multiplication ( 數 ) Proposition 3.1.9. r∈ R, A ∈ Mm×n, B∈ Mn×l.

r(AB) = (rA)B = A(rB).

3.1. 45

Proof. A = [a_{i j}], B = [b_{j k}], 1≤ i ≤ m, 1 ≤ j ≤ n 1≤ k ≤ l. r(AB) k-th column r AB k-th column,

r(b_{1 k}a₁+ b_{2 k}a₂+··· + bn ka_n). (3.11) 1≤ j ≤ n , (rA) j-th column ra_j. (rA)B k-th column

b1 k(ra1) + b2 k(ra2) +··· + bn k(ran). (3.12) rB k-th column





 rb_1k rb2k

... rb_nk





,

A(rB) k-th column

(rb_{1 k})a₁+ (rb_{2 k})a₂+··· + (rbn k)a_n. (3.13) Proposition 1.2.3 性 , (3.11), (3.12), (3.13)

. 

Proposition 3.1.8 Proposition 3.1.9 , 性

column . row

, transpose ( ) .

, 性 ( (AB)C = A(BC)).

, 大 . 學 row

, . 數

性 , . A B , B A,

A∈ M2×3, B∈ M3×4 . A B B A ,

數 , AB̸= BA, A∈ M2×3, B∈ M3×2 . A, B

, AB BA 數 . AB̸= BA,

A = [a b

c d ]

, B = [1 0

0 −1 ]

, AB =

[a −b c −d ]

, BA =

[ a b

−c −d ]

,

b = c = 0 , AB = BA. .

A, B Proposition 3.1.8 Proposition 3.1.9 (A− B)(A + B) = A²− AB + BA − B², AB̸= BA, (A− B)(A + B) = A²− B².

, . zero

matrix ( ) O ( O = [a_{i, j}] entry a_{i, j}= 0). O n× n

square matrix, A∈ Mn×n, OA = AO = O. identity

matrix. n× n identity matrix, I_n . I_n i-th column e_i,

e₁, e₂, . . . , e_n Rⁿ standard basis ( column vector).

I₃=



1 0 0 0 1 0 0 0 1



,I₄=







1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1





.

, A∈ Mm×n, B∈ Mn×l AIn= A, InB = B.

, A n× n matrix, AI_n= I_nA = A.

Question 3.2. A∈ Mn×n, (A− 2In)²= A²− 4A + 4In ?

Question 3.3. In n× n A∈ Mm×n AIn= A.

n× n square matrix (i, i)-th entry diagonal entry. diagonal entries , entry 0, diagonal matrix. Identity matrix

diagonal matrix. diagonal entry 1, entry 0. ,

r∈ R, rIn diagonal matrix. diagonal entry r, entry 0.

A∈ Mm×n, B∈ Mn×l rA = A(rI_n), rB = (rI_n)B.

Question 3.4. Proposition 3.1.9 n× n square matrix A, (rI_n)A = A(rI_n).

, n× n diagonal matrix n× n square matrix . 前

[1 0 0 −1

]

2× 2 .

3.2. Transpose Operation

transpose ( ) ,

row . column row .

Definition 3.2.1. m× n matrix A = [ai j]. A transpose n× m, A^T, 1≤ i ≤ n, 1 ≤ j ≤ m, A^T (i, j)-th entry A ( j, i)-th entry.

well-defined, m× n matrix A,

n× m matrix A^T. 1≤ i ≤ n, 1 ≤ j ≤ m, A^T i-th row

j-th column 數 A j-th row i-th column 數. A row

column 數 m n, j, i 1≤ j ≤ m, 1 ≤ i ≤ n,

A entry A^T i-th row j-th column . .

. Example 3.2.2.

A =

[ 1 2 3

−1 −2 −3 ]

,

3.2. Transpose Operation 47

A^T 3× 2 matrix.

A^T=



x1 1 x1 2

x2 1 x2 2

x3 1 x3 2



.

A^T (1, 1)-th entry x1 1 A (1, 1)-th entry, 1. A^T (2, 1)-th entry x2 1

A (1, 2)-th entry, 2, A^T (3, 1)-th entry x_{3 1} A (1, 3)-th entry, 3.

x1 2=−1, x2 2=−2, x3 2=−3, A^T=



1 −1 2 −2 3 −3



.

A^T 1-st column A 1-st row column , A^T 2-nd column A 2-nd row column . A^T 1-st, 2-nd 3-rd row A 1-st, 2-nd 3-rd column row .

row column, A A^T

row column .

Lemma 3.2.3. A m× n matrix. 1≤ i ≤ n, A^T i-th row A i-th column row vector. 1≤ j ≤ m, A^T j-th column A j-th row column vector.

Proof. A^T n×m matrix. A = [a_{k l}], A^T= [a^′_{i j}]. 1≤ i ≤ n, A^T i-th row [a^′_{i 1} a^′_{i 2} ··· a^′_{i m}]

. a^′_{i 1}= a_{1 i}, a^′_{i 2}= a_{2 i}, . . . , a^′_{i m}= a_{m i}, A^T i-th row

[a_{1 i} a_{2 i} ··· am i

],

A i-th column 



 a_{1 i} a_{2 i} ... a_{m i}







row vector . , 1≤ j ≤ m, A^T j-th column A j-th

row column vector. 

transpose 性 .

Proposition 3.2.4. A, B m× n matrix, C n× l matrix. 性 . (1) (A^T)^T= A.

(2) (A + B)^T= A^T+ B^T. (3) (AC)^T= C^TA^T.

Proof. A^T n× m matrix, (A^T)^T m× n matrix, A 數 . , A^T+ B^T n× m matrix (A + B)^T 數 . C^T l× n matrix, C^TA^T l× m matrix. AC m× l matrix, (AC)^T l× m matrix C^TA^T 數 .

(1) (A^T)^T A m×n matrix, 1≤ i ≤ n, (A^T)^T i-th column A i-th column. (A^T)^T i-th column Lemma 3.2.3 A^T i-th row column vector, A^T i-th row A i-th column. (A^T)^T= A.

(2) A^T+ B^T (A + B)^T n× m matrix, 1≤ i ≤ m, A^T+ B^T i-th column (A + B)^T i-th column. A^T+ B^T i-th column A^T B^T i-th column . Lemma 3.2.3 A B i-th row . , (A + B)^T i-th column A + B i-th row, A B i-th row . (A + B)^T= A^T+ B^T.

(3) (AC)^T column AC row , A C row

, entry . 1≤ i ≤ l, 1 ≤ j ≤ m, (AC)^T

(i, j)-th entry AC ( j, i)-th entry, A j-th row C i-th column ( Rⁿ ) . , C^TA^T (i, j)-th entry C^T i-th row A^T j-th column

. C i-th column A j-th row , (AC)^T= C^TA^T. 

Question 3.5. A m× n matrix, r ∈ R. (rA)^T= rA^T.

n× n square matrix, A^T= A, A symmetric matrix.

diagonal matrix symmetric matrix. 學 symmetric matrix

性 , symmetric matrix .

Corollary 3.2.5. A n× n square matrix, B m× n matrix. symmetric matrix.

A + A^T, BB^T, B^TB.

Proof. Proposition 3.2.4, (A + A^T)^T= A^T+ (A^T)^T= A^T+ A, A + A^T sym- metric matrix. , (BB^T)^T= (B^T)^TB^T= BB^T, BB^T symmetric matrix.

B^TB symmetric matrix. 

Proposition 3.2.4, row . 1× m

matrix m× n matrix . A∈ M1×m, B∈ Mm×n,

A =[

a₁ a₂ ··· am

], B =







b1 1 b1 2 ··· b1 n

b2 1 b2 2 ··· b2 n

... ... ··· ... bm 1 bm 2 ··· bm n





.

(AB)^T= B^TA^T, column vector

(AB)^T=







b_{1 1} b_{2 1} ··· bm 1

b_{1 2} b_{2 2} ··· bm 2

... ... ··· ... b_{1 n} b_{2 n} ··· bm n











 a₁ a₂ ... a_m





= a1





 b_{1 1} b_{1 2} ... b_{1 n}





+ a2





 b_{2 1} b_{2 2} ... b_{2 n}





+··· + am





 b_{m 1} b_{m 2} ... b_{m n}





.

3.2. Transpose Operation 49

(AB)^T= a₁(₁b)^T+ a₂(₂b)^T+··· + am(_mb)^T, (_ib)^T B i-th row ( column ). Proposition 3.2.4 (AB)^T

AB = a₁(₁b) + a₂(₂b) +··· + am(_mb).

[a1 ··· am

]





b1 1 ··· b1 n

... ··· ... b_{m 1} ··· bm n



 = a¹[

b1 1 ··· b1 n

]+··· + am

[bm 1 ··· bm n

] (3.14)

, A = [ai j] m× n matrix B = [b_jk] n× l matrix.

(AB)^T= B^TA^T. B^TA^T i-th column, B^T A^T i-th column.

A^T i-th column, A i-th row , (ia)^T. (AB)^T i-th column B^T(_ia)^T. Proposition 3.2.4 , AB i-th row

(B^T(_ia)^T)^T= ((_ia)^T)^T(B^T)^T=_iaB.

言 ,







— ₁a —

— ₂a — ...

— _ma —





B =







— ₁a B —

— ₂a B — ...

— _ma B —





. (3.15)

(3.14), .

Proposition 3.2.6. A = [a_{i j}] m× n matrix B = [b_jk] n× l matrix, 1≤ i ≤ m, AB i-th row

ia B =[

ai 1 ··· ai n

]





b1 1 ··· b1 l

... ··· ... b_{n 1} ··· bn l



 = a^{i 1}[

b1 1 ··· b1 l

]+··· + ai n

[bn 1 ··· bn l

].

.

Proposition 3.2.7. A∈ Mm×n, B∈ Mn×l,C∈ Ml×k, (AB)C = A(BC).

Proof. AB∈ Mm×l, (AB)C∈ Mm×k. BC∈ Mn×k, A(BC)∈ Mm×k (AB)C . 1≤ i ≤ m, 1 ≤ j ≤ k, (AB)C A(BC) (i, j)-th entry . _ia A i-th row, c_j C j-th column. (AB)C (i, j)-th entry AB i-th row C j-th column. Proposition 3.2.6 AB i-th row _ia B, (AB)C (i, j)-th entry (_ia B)c_j. , A(BC) (i, j)-th entry A i-th row BC j-th column. Definition 3.1.6 BC j-th row Bcj, A(BC) (i, j)-th entry ia (Bc_j).

, (AB)C A(BC) (i, j)-th entry , (_ia B)c_j=_ia (Bc_j).

ia c_j row column, , _ia c_j .

ia =[

a₁ a₂ ··· an

], B =







b1 1 b1 2 ··· b1 l

b_{2 1} b_{2 2} ··· b2 l

... ... ··· ... bn 1 bn 2 ··· bn l





, c_j=





 c1

c₂ ... c_l







ia B =[

a₁ a₂ ··· an

]







b1 1 b1 2 ··· b1 l

b2 1 b2 2 ··· b2 l

... ... ··· ... bn 1 bn 2 ··· bn l





=[

ia b_{1 i}a b₂ ··· ia b_l]

(_ia B)c_j=[

ia b_{1 i}a b₂ ··· ia b_l]





 c1

c2

... cl





= c₁(_iab₁) + c₂(_iab₂) +··· + cl(_iab_l).

, (Definition 3.1.4)

Bc_j=







b1 1 b1 2 ··· b1 l

b2 1 b2 2 ··· b2 l

... ... ··· ... bn 1 bn 2 ··· bn l











 c1

c2

... cl





= c₁b₁+ c₂b₂+··· + clb_l,

性 (Proposition 1.4.2),

ia (Bc_j) =_ia(c₁b₁+ c₂b₂+··· + clb_l) = c₁(_iab₁) + c₂(_iab₂) +··· + cl(_iab_l).

(_ia B)c_j=_ia (Bc_j), (AB)C = A(BC). 

Question 3.6. A∈ Mm×n, r∈ R, (rI_m)A = rA = A(rI_n) Proposition 3.2.7 Proposition 3.1.9.

(Proposition 3.2.7), , ,

ABC . , A , (AA)A = A(AA),

A³ . , n A , Aⁿ .

3.3. Elementary Matrix

A E, E row A . ,

A elementary row operation, A .

elementary matrix.

A = [a_{i j}] m× n matrix. identity matrix I_m A . I_m i-th

row [

0 ··· 1 ··· 0] ˆi

3.3. Elementary Matrix 51

i-th entry 1, entry 0. Proposition 3.2.6, I_mA i-th row [ 0 ··· 1 ··· 0 ]

A = 0₁a +··· + 1ia +··· + 0ma =_ia,

( 1 A i-th row, 0 A row , A i-th row.)

言 , Im A , A row , ImA = A. j̸= i

E I_m i-th row j-th entry 1 entry 0, i-th row row

. EA i-th row A j-th row, EA A i-th

row A j-th row, row .

i-th row j-th row elementary row operation Im E,

E =













 1

. ..

0 1

. ..

1 0

. ..

1















(3.16)

前 , EA i-th row A j-th row, EA j-th row A i-th row,

row . 言 , EA A i-th row j-th row elementary

row operation .

Im i-th row 數 r Im j-th row E,

E =













 1

. ..

1 . ..

r 1

. ..

1















(3.17)

E j-th row i-th entry r, j-th entry 1. Proposition 3.2.6, EA j-th

row r A i-th row A j-th row, row . 言 ,

EA A i-th row 數 r A j-th row elementary row operation

.

Im i-th row 數 r E,

E =













 1

. ..

1 r

1 . ..

1















(3.18)

EA i-th row A i-th row r, row . ,

EA A i-th row 數 r elementary row operation

.

, m× n matrix elementary row operation,

m×m matrix. m×m matrix m×m identity matrix

Im elementary row operation . elementary

matrix. (3.16), (3.17), (3.18) elementary matrix .

m× n matrix A, elementary row operations, A elementary matrix. elementary row operations, A elementary row operation elementary matrix E1.

E₁A elementary row operation elementary matrix

E2. E2(E₁A) A elementary row operations .

, E2(E1A) (E2E1)A. , A

elementary row operations, A ,

elementary row operations elementary matrices . ,

elementary matrices , elementary matrices

.

Question 3.7. elementary matrices .

elementary row operations,

. 前 elementary matrices ,

. “clever” , elementary

row operation . , m× n matrix

A elementary row operations, augmented matrix [A|Im].

m× (n + m) , n columns ( 前 n columns) A, m

columns ( m column) Im. A elementary row operation, elementary matrix E1, [A|Im] elementary row operation ,

E₁[A|Im].

E1[A|Im] = [E₁A|E1Im] = [E₁A|E1].

, [A|Im] elementary row operation,

A elementary row operation , elementary row

operation elementary matrix. elementary row operation, elementary row operation elementary matrix E₂, elementary row operation [E₁A|E1] E₂[E₁A|E1] = [E₂(E₁A)|E2E₁]. , [A|Im] elementary row operations , [A^′|E],

A^′ A elementary row operations ,

3.3. Elementary Matrix 53

E elementary row operations elementary matrices

, EA = A^′. .

Lemma 3.3.1. A m× n matrix. A elementary row operations

A^′, m× m matrix E EA = A^′, E elementary row

operations elementary matrix . augmented

matrix [A|Im] elementary row operations , augmented matrix [A^′|E].

Example 3.3.2.

A =



 2 −4 4 −6 1 −2 1 −1 4 −8 4 −4





reduced echelon form, elementary matrices E EA reduced echelon form.

augmented matrix

[A|I3] =



 2 −4 4 −6 1 0 0 1 −2 1 −1 0 1 0 4 −8 4 −4 0 0 1





augmented matrix 1-st 2-nd row ,



 1 −2 1 −1 0 1 0 2 −4 4 −6 1 0 0 4 −8 4 −4 0 0 1





augmented matrix 1-st row −2 2-nd row ,



 1 −2 1 −1 0 1 0 0 0 2 −4 1 −2 0 4 −8 4 −4 0 0 1



.

augmented matrix 1-st row −4 3-rd row



 1 −2 1 −1 0 1 0 0 0 2 −4 1 −2 0

0 0 0 0 0 −4 1



.

augmented matrix 2-nd row 1/2



 1 −2 1 −1 0 1 0 0 0 1 −2 ¹₂ −1 0

0 0 0 0 0 −4 1



.

augmented matrix 2-nd row −1 1-st row



 1 −2 0 1 −¹₂ 2 0 0 0 1 −2 ¹₂ −1 0

0 0 0 0 0 −4 1



.

augmented matrix [A^′|E], A reduced echelon form A^′

EA. ,

EA =



 −¹₂ 2 0

1

2 −1 0

0 −4 1







 2 −4 4 −6 1 −2 1 −1 4 −8 4 −4



 =



 1 −2 0 1

0 0 1 −2

0 0 0 0



.

E elementary row operations elementary matrices . A 3× 4 matrix, elementary row operation elementary matrix E1 3×3 identity matrix I3 1-st 2-nd row , elementary matrix E₂ I₃ 1-st row −2 2-nd row . elementary matrix E₃

I3 1-st row −4 3-rd row . elementary matrix E4 I3 2-nd row 1/2, elementary matrix E₅ I3 2-nd row −1 1-st row

. ,

E1=



 0 1 0 1 0 0 0 0 1



, E₂=



 1 0 0

−2 1 0 0 0 1



, E₃=



 1 0 0

0 1 0

−4 0 1



,

E4=



 1 0 0 0 ¹₂ 0 0 0 1



, E₅=



 1 −1 0

0 1 0

0 0 1





elementary matrices ,

E₅E4E3E2E1=



 −¹₂ 2 0

1

2 −1 0

0 −4 1



 = E.

elementary matrices 性 .

elementary matrix, elementary column operation.

前 , column (Definition 3.1.6) .

E identity In i-th row j-th row elementary matrix,

E I_n i-th column j-th column . E m× n matrix A

, AE, A i-th column j-th column . , E

identity In i-th row 數 r elementary matrix, E

I_n i-th column r. E m× n matrix A , AE,

A i-th column r. In i-th row r

j-th row elementary matrix E. elementary matrix column operation, In j-th column r i-th column ( i-th column r j-th column

). E m× n matrix A , AE, A j-th

column r i-th column . Example 3.3.3.

E₁=



 1 0 0 0 0 1 0 1 0



, E₂=



 10 0 0 0 1 0 0 0 1



, E₃=



 1 0 10 0 1 0 0 0 1



, A =



 1 2 3

−1 −2 −3 11 22 33





3.4. Matrix System of Linear Equations 55

E₁ I₃ 2-nd row 3-rd row , I₃ 2-nd column 3-rd column .

E1A =



 1 0 0 0 0 1 0 1 0







 1 2 3

−1 −2 −3 11 22 33



 =



 1 2 3

11 22 33

−1 −2 −3



,

AE₁=



 1 2 3

−1 −2 −3 11 22 33







 1 0 0 0 0 1 0 1 0



 =



 1 3 2

−1 −3 −2 11 33 22



.

E2 I3 1-st row 10, I3 1-st column 10.

E₂A =



 10 0 0 0 1 0 0 0 1







 1 2 3

−1 −2 −3 11 22 33



 =



 10 20 30

−1 −2 −3 11 22 33



,

AE2=



 1 2 3

−1 −2 −3 11 22 33







 10 0 0 0 1 0 0 0 1



 =



 10 2 3

−10 −2 −3 110 22 33



.

E₃ I₃ 3-rd row 10 1-st row, I₃ 1-st column 10

3-rd column.

E3A =



 1 0 10 0 1 0 0 0 1







 1 2 3

−1 −2 −3 11 22 33



 =



 111 222 333

−1 −2 −3 11 22 33



,

AE₁=



 1 2 3

−1 −2 −3 11 22 33







 1 0 10 0 1 0 0 0 1



 =



 1 2 13

−1 −2 −13 11 22 143



.

3.4. Matrix System of Linear Equations

, rank

. ,

步 .

, Rⁿ ,

row vector, column vector . n× 1 matrix.

u, v∈ Rⁿ, u^Tv u, v ( u· v). ,

a₁₁x₁ + a₁₂x₂ + ··· + a1nx_n = b₁ a₂₁x₁ + a₂₂x₂ + ··· + a2nx_n = b₂

...

a_m1x₁ + a_m2x₂ + ··· + amnx_n = b_m

A =







a₁₁ a₁₂ ··· a1n

a₂₁ a₂₂ ··· a2n

... ... ... ... a_m1 a_m2 ··· amn





, x =





 x₁ x₂ ... x_n





, b =





 b₁ b₂ ... b_m





,

Ax = b . x₁= c₁, x₂= c₂, . . . , x_n= c_n, ,

c =





 c₁ c2

... cm





,

, x = c∈ Rⁿ Ax = b . A

m× n matrix c n× 1 matrix b m× 1 matrix, Ac = b.

3.4.1. 性. m× n matrix A b∈ R^m,

Ax = b .

b∈ R^m Ax = b . c =



 c₁

... c_n



 , Ac = b.

c1a₁+··· + cna_n= b,

a₁, . . . , an A column vectors. , b A column vectors linear combination. b∈ Span(a1, . . . , a_n). , b∈ Span(a1, . . . , a_n),

c1, . . . , c_n∈ R b = c₁a₁+··· + cna_n. x1= c₁, . . . , x_n= c_n Ax = b . 性 .

Lemma 3.4.1. A∈ Mm×n b∈ R^m. Ax = b b∈ Span(a1, . . . , a_n), a₁, . . . , an A column vectors.

m× n matrix A b∈ R^m, Ax = b

. 學 , .

A column vectors R^m , Span(a1, . . . , an)⊆ R^m. Lemma 3.4.1 , b∈ R^m Ax = b , b∈ R^m b∈ Span(a1, . . . , a_n).

Span(a1, . . . , a_n) =R^m. , Span(a1, . . . , a_n) =R^m, b∈ R^m b∈ Span(a1, . . . , an). Lemma 3.4.1 b∈ R^m Ax = b .

, b∈ R^m, Ax = b Span(a₁, . . . , a_n) =R^m .

Ax = ei, e₁, . . . , e_m∈ R^m R^m standard basis.

b∈ R^m, Ax = b , i = 1, . . . , m,

c_i∈ Rⁿ x = c_i Ax = e_i . i = 1, . . . , m

Aci= ei. n× m matrix C, i-th column c_i.

AC = A



   c₁ c₂ ··· cm



 =



   

Ac1 Ac2 ··· Acm



 =



   e₁ e₂ ··· em



 = I_m.