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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

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## 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 = [ai 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 = [ai j], a1 1= 1, a1 2= 0, a1 3= 2, a1 4= 3,

a2 1= 0, a2 2= 1, a2 3= 5, a2 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 aj .

(3.1) A,

1a =[

1 0 2 3] , 2a[

0 1 5 8]

, 3a =[

2 1 1 0]

39

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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 = [ai j] m× n matrix A= [ai j] m× n matrix.

A = A m = m, n = n 1≤ i ≤ m 1≤ j ≤ n ai j= ai j.

. Rn

, Rn 數 .

Mm×n , Mm×n

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 ci j= ai j+ bi j. 數 r, rA = [di j]

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

A =





a1 1 a1 2 ··· a1 n

a2 1 a2 2 ··· a2 n

... ... . .. ... am 1 am 2 ··· am n



, B =





b1 1 b1 2 ··· b1 n

b2 1 b2 2 ··· b2 n

... ... . .. ... bm 1 bm 2 ··· bm n





A + B =





a1 1+ b1 1 a1 2+ b1 2 ··· a1 n+ b1 n a2 1+ b2 1 a2 2+ b2 2 ··· a2 n+ b2 n

... ... . .. ... am 1+ bm 1 am 2+ bm 2 ··· am n+ bm n





rA =





ra1 1 ra1 2 ··· ra1 n

ra2 1 ra2 2 ··· ra2 n

... ... . .. ... ram 1 ram 2 ··· ram n





(5)

3.1. 41

Proposition 3.1.3. Mm×n ,:

(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 = [bj] n× 1 matrix ( Rn column vector). ai A i-th column

Ab =





a1 1 a1 2 ··· a1 n

a2 1 a2 2 ··· a2 n

... ... . .. ... am 1 am 2 ··· am n







 b1 b2 ... bn



= b1a1+ b2a2+··· + bnan.

(6)

, A∈ Mm×n column 數 n b∈ Mn×1 row 數 n,

Ab Ab m× 1 matrix ( Rm column vector). column

vector,

Ab = b1



 a11

a21 ... am1



+ b2



 a12

a22 ... am2



+··· + bn



 a1n

a2n ... amn



=





b1a11+ b2a12+··· + bna1n

b1a21+ b2a22+··· + bna2n ...

b1am1+ b2am2+··· + bnamn



. (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+··· + bnan. (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 bk∈ Mn×1, 1≤ k ≤ l,

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

A

 b1 b2 ··· bl

 =

Ab1 Ab2 ··· Abl

Abk m× 1 matrix, AB m× l matrix. .

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3.1. 43

Deﬁnition 3.1.6. A = [ai j] m× n matrix B = [bjk] n× l matrix, AB = C = [cik] m× l matrix, 1≤ k ≤ l, C k-th column ck

ck= Abk=





a1 1 a1 2 ··· a1 n

a2 1 a2 2 ··· a2 n

... ... . .. ... am 1 am 2 ··· am n







 b1 k

b2 k ... bn k



= b1 ka1+ b2 ka2+··· + bn kan. (3.5)

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

Abk) i entry. (3.4) A i-th row ia B k-th

column bk Rn . 言 , AB = [ci k], AB (i, k)-th entry ci k ci k=ia bk= ai 1b1 k+ ai 2b2 k+··· + ai nbn k=

### ∑

n j=1

ai jbj 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 ?

.

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

(2) (A + A)B = AB + AB.

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Proof. A + A m× n B + B n× l , 數 . A = [ai j], A= [ai j], B = [bj k], B = [bj 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 bk+ bk, B k-th column B k-th column.

A(B + B) k-th column

A(bk+ bk) = A





b1k+ b1k b2k+ b2k

... bnk+ bnk



= (b1k+ b1k)a1+ (b2k+ b2k)a2+··· + (bnk+ bnk)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



 b1k b2k ... bnk



= (b1ka1+ b2ka2+··· + bnkan) + (b1ka1+ b2ka2+··· + bnkan). (3.8)

Proposition 1.2.3 性 , (3.7) (3.8) .

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

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

, 1≤ j ≤ n , A + A j-th column aj+ aj, A j-th column A j-th column. (A + A)B k-th column

(A + A)bk= (A + A)



 b1k b2k ... bnk



= b1k(a1+ a1) + b2k(a2+ a2) +··· + bnk(an+ an). (3.9)

, AB + AB k-th column AB k-th column AB k-th column, AB + AB k-th column

A



 b1k b2k

... bnk



+ A



 b1k b2k

... bnk



= (b1ka1+ b2ka2+··· + bnkan) + (b1ka1+ b2ka2+··· + bnkan). (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).

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3.1. 45

Proof. A = [ai j], B = [bj k], 1≤ i ≤ m, 1 ≤ j ≤ n 1≤ k ≤ l. r(AB) k-th column r AB k-th column,

r(b1 ka1+ b2 ka2+··· + bn kan). (3.11) 1≤ j ≤ n , (rA) j-th column raj. (rA)B k-th column

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



 rb1k rb2k

... rbnk



,

A(rB) k-th column

(rb1 k)a1+ (rb2 k)a2+··· + (rbn k)an. (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

A∈ M2×3, B∈ M3×4 . A B B A ,

, 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 = [ai, j] entry ai, j= 0). O n× n

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

matrix. n× n identity matrix, In . In i-th column ei,

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e1, e2, . . . , en 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, AIn= InA = 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(rIn), rB = (rIn)B.

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

, 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, AT, 1≤ i ≤ n, 1 ≤ j ≤ m, AT (i, j)-th entry A ( j, i)-th entry.

well-deﬁned, m× n matrix A,

n× m matrix AT. 1≤ i ≤ n, 1 ≤ j ≤ m, AT i-th row

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

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

A entry AT i-th row j-th column . .

. Example 3.2.2.

A =

[ 1 2 3

−1 −2 −3 ]

,

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3.2. Transpose Operation 47

AT 3× 2 matrix.

AT=

x1 1 x1 2

x2 1 x2 2

x3 1 x3 2

.

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

A (1, 2)-th entry, 2, AT (3, 1)-th entry x3 1 A (1, 3)-th entry, 3.

x1 2=−1, x2 2=−2, x3 2=−3, AT=

1 −1 2 −2 3 −3

.

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

row column, A AT

row column .

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

Proof. AT n×m matrix. A = [ak l], AT= [ai j]. 1≤ i ≤ n, AT i-th row [ai 1 ai 2 ··· ai m]

. ai 1= a1 i, ai 2= a2 i, . . . , ai m= am i, AT i-th row

[a1 i a2 i ··· am i

],

A i-th column



 a1 i a2 i ... am i





row vector . , 1≤ j ≤ m, AT 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) (AT)T= A.

(2) (A + B)T= AT+ BT. (3) (AC)T= CTAT.

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Proof. AT n× m matrix, (AT)T m× n matrix, A 數 . , AT+ BT n× m matrix (A + B)T 數 . CT l× n matrix, CTAT l× m matrix. AC m× l matrix, (AC)T l× m matrix CTAT 數 .

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

(2) AT+ BT (A + B)T n× m matrix, 1≤ i ≤ m, AT+ BT i-th column (A + B)T i-th column. AT+ BT i-th column AT BT 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= AT+ BT.

(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 ) . , CTAT (i, j)-th entry CT i-th row AT j-th column

. C i-th column A j-th row , (AC)T= CTAT. 

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

n× n square matrix, AT= A, A symmetric matrix.

diagonal matrix symmetric matrix. 學 symmetric matrix

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

A + AT, BBT, BTB.

Proof. Proposition 3.2.4, (A + AT)T= AT+ (AT)T= AT+ A, A + AT sym- metric matrix. , (BBT)T= (BT)TBT= BBT, BBT symmetric matrix.

BTB 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= BTAT, column vector

(AB)T=





b1 1 b2 1 ··· bm 1

b1 2 b2 2 ··· bm 2

... ... ··· ... b1 n b2 n ··· bm n







 a1 a2 ... am



= a1



 b1 1 b1 2 ... b1 n



+ a2



 b2 1 b2 2 ... b2 n



+··· + am



 bm 1 bm 2 ... bm n



.

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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

... ··· ... bm 1 ··· bm n

 = a1[

b1 1 ··· b1 n

]+··· + am

[bm 1 ··· bm n

] (3.14)

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

(AB)T= BTAT. BTAT i-th column, BT AT i-th column.

AT i-th column, A i-th row , (ia)T. (AB)T i-th column BT(ia)T. Proposition 3.2.4 , AB i-th row

(BT(ia)T)T= ((ia)T)T(BT)T=iaB.





1a —

2a — ...

ma —



B =





1a B

2a B — ...

ma B



. (3.15)

(3.14), .

Proposition 3.2.6. A = [ai j] m× n matrix B = [bjk] n× l matrix, 1≤ i ≤ m, AB i-th row

ia B =[

ai 1 ··· ai n

]



b1 1 ··· b1 l

... ··· ... bn 1 ··· bn l

 = ai 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, cj 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)cj. , 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 (Bcj).

, (AB)C A(BC) (i, j)-th entry , (ia B)cj=ia (Bcj).

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ia cj row column, , ia cj .

ia =[

a1 a2 ··· an

], B =





b1 1 b1 2 ··· b1 l

b2 1 b2 2 ··· b2 l

... ... ··· ... bn 1 bn 2 ··· bn l



, cj=



 c1

c2 ... cl





ia B =[

a1 a2 ··· an

]





b1 1 b1 2 ··· b1 l

b2 1 b2 2 ··· b2 l

... ... ··· ... bn 1 bn 2 ··· bn l



=[

ia b1 ia b2 ··· ia bl]

(ia B)cj=[

ia b1 ia b2 ··· ia bl]



 c1

c2

... cl



= c1(iab1) + c2(iab2) +··· + cl(iabl).

, (Deﬁnition 3.1.4)

Bcj=





b1 1 b1 2 ··· b1 l

b2 1 b2 2 ··· b2 l

... ... ··· ... bn 1 bn 2 ··· bn l







 c1

c2

... cl



= c1b1+ c2b2+··· + clbl,

ia (Bcj) =ia(c1b1+ c2b2+··· + clbl) = c1(iab1) + c2(iab2) +··· + cl(iabl).

(ia B)cj=ia (Bcj), (AB)C = A(BC). 

Question 3.6. A∈ Mm×n, r∈ R, (rIm)A = rA = A(rIn) 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 = [ai j] m× n matrix. identity matrix Im A . Im i-th

row [

0 ··· 1 ··· 0] ˆi

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3.3. Elementary Matrix 51

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

A = 01a +··· + 1ia +··· + 0ma =ia,

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

E Im 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)

, 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)

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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 ,

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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

.

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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.

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

E In 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

In 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

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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

. ,

, Rn ,

row vector, column vector . n× 1 matrix.

u, v∈ Rn, uTv u, v ( u· v). ,

a11x1 + a12x2 + ··· + a1nxn = b1 a21x1 + a22x2 + ··· + a2nxn = b2

...

am1x1 + am2x2 + ··· + amnxn = bm

A =





a11 a12 ··· a1n

a21 a22 ··· a2n

... ... ... ... am1 am2 ··· amn



, x =



 x1 x2 ... xn



, b =



 b1 b2 ... bm



,

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Ax = b . x1= c1, x2= c2, . . . , xn= cn, ,

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∈ Rm,

Ax = b .

b∈ Rm Ax = b . c =

 c1

... cn

 , Ac = b.

c1a1+··· + cnan= b,

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

c1, . . . , cn∈ R b = c1a1+··· + cnan. x1= c1, . . . , xn= cn Ax = b . 性 .

Lemma 3.4.1. A∈ Mm×n b∈ Rm. Ax = b b∈ Span(a1, . . . , an), a1, . . . , an A column vectors.

m× n matrix A b∈ Rm, Ax = b

. 學 , .

A column vectors Rm , Span(a1, . . . , an)⊆ Rm. Lemma 3.4.1 , b∈ Rm Ax = b , b∈ Rm b∈ Span(a1, . . . , an).

Span(a1, . . . , an) =Rm. , Span(a1, . . . , an) =Rm, b∈ Rm b∈ Span(a1, . . . , an). Lemma 3.4.1 b∈ Rm Ax = b .

, b∈ Rm, Ax = b Span(a1, . . . , an) =Rm .

Ax = ei, e1, . . . , em∈ Rm Rm standard basis.

b∈ Rm, Ax = b , i = 1, . . . , m,

ci∈ Rn x = ci Ax = ei . i = 1, . . . , m

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

AC = A

 c1 c2 ··· cm

 =

Ac1 Ac2 ··· Acm

 =

 e1 e2 ··· em

 = Im.

[r]

elementary row operations reduced echelon form,. echelon form Gauss

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