线性代数笔记第三章
时间：2025-07-07
同济大学
3
1.
1.1.(A,b),
Ax=b.
1.2.
(i)—–Gauss
(ii)
—–
1.3.
(1)ri rj;(2)ri×k(k=0);(3)ri+krj.
1.4.1.5.
(1)ri rjri rj(2)ri×k
r1;
i×
+
(
)
同济大学
(2)
1
0.
0··· 0··· 0
··· . .. 0···
...0
···
1.6.
(Step0)(Step1)(Step2)(Step3)1.7.1.8.
(
(Step1)(Step2)(Step3)2.
2.1.
(
(1)En(2)En(3)En
(
01 0 0 0
00···1
0 0
00
···0···
1
............
....
.. 00···0···0···
1 ................ .. 00
···
0···
0···
···
GameOver
0;
(Step0).
) ErOO
O
,r≥0.
)
ri rj,En(i,j);
=0),
ri×k(kEn(i(k));
ri+krj,
En(ij(k)).
(
(
),
2
).)
同济大学
2.2.
(
-
)
(1)(2)
(1)A
1
βT1
.. . ...
βT1
...
0
1
βTβT .. .j. . . .i . . 1
βT βT .. . .i. . 1
.j. . = βTn
βTn
(2)
A
1
...
1
α1···αi···αj···αm
.. . =
α
1···αi···αj+kαi···
αm
.
2.3.
A
A
“ =”“= ”
A
F
E.
A=L1L2···LlFR1R2···Rr
F
|F|=0,
|A|=0.
3
k
1
.. . 1
A=P1P2···PN.
同济大学
2.4.
+
(
A →B)
A
B
P,
PA=B;
(
A →B)
A
B
Q,
AQ=B;
(
A →B)
A
B
P,Q,
2.5.r
A～E A～E A～c
A
E.2.6.
(A,E)～r
(E,A 1)
∵A 1(A,E)=(E,A 1).
(A,E)
E
2.7.
AX=B,AX=b.
X=A 1B,(A,B)～r
(E,A 1B)
3.
3.1.
(
)
A
=A=
3.2.
Am×n
k
k
k
kk
k
=CmkCnk.
01234567
89012345
67890123
45678901
23
4
5
67
89
5×8
3.3.
(rank)
4
PAQ=B.
A
同济大学
(i)R(A)=
A
(ii)R(O)=0.3.4.
(1)(2)(3.5.
r
(r+1)
(
)
)
(i)
s
= R(A)≥s;
t
= R(A)<t.
(ii)R(Am×n)≤min{m,n}.(iii)R(A)=R(AT).
(iv)R(A)=R( A),R(A)=R(λA),λ=0.(v)An×n(vi)R(A3.6.
R(A)=n A
)≤R(A).
()
000000
920000000000
370000
4023000000003
67
3
40
53
00
00 00
6×8
(
)
3.7.
A →B,R(A)≥R(B)R(A)=r
B
(r+1)
(a)ri rj,(b)ri×k,
B(c)ri+krj
ij,
.. .
ri+krj
. . .
ij,
=
5
j .. . ri +k . . .
ii ,
...
rj =0+0. . ..
j
同济大学
3.8.
(
R(PAQ)=R(A),P,Q
).
(1)(2)3.9.
A,B∈Mm×n.R(A)=R(B) A～B.
3.10.
(
)32050
A= 3 236 1 2015 3 A
16 4 14
(Step1)
A 4
A row
→B=
16 4 10 431 1
0
004 8 00
00
R(A)=R(B)=B
=3.
(Step2)
B
1,2,4
(Step3)
A
A′=
33′
21A3.11.
12 11
A= (Step1)
3
2a 1
a,b
5
6
b
A
,312 1A→ 1
0 4a+3 4
005 ab 1
6
25
26 05
6 1
A
同济大学
(Step2)
(i)a=5,b=1
R(A)=2;
(ii)(a 5)(b 1)=0R(A)=3.
4.
4.1.max{R(A),R(B)}≤R(A,B)≤R(A)+R(B).
(A,B)←→(A′,B′)
column
4.2.R(A+B)≤R(A)+R(B)
R(A+B)≤R(A+B,B),(A+B,B)←→(A,B).
column
4.3.R(AB)≤min{R(A),R(B)}
(i)
A
R(AB)≤R(A).
PA=
A′O
,
A′
R(A′)=R(A).
R(AB)=R(PAB)=R≤A′B(ii)
=A′
A
′
O
B
=R
ABO
′
=R(A′B)
=R(A)
R(AB)≤R(B)
R(AB)=R((AB)T)=R(BTAT)≤R(BT)=R(B).
4.4.
(i)(ii)
P,Q
R(A)=R(PA)=R(AQ)=R(PAQ).
Am×nBn×l=Cm×l,R(A)=n,R(B)=R(C).R(A)=R(C).
(iii)
Am×nBn×l=Cm×l,R(B)=n,
= AP,
(ii)Am×n
EnO
.
B=
BO ,
PC=PAB=BO =R(B).
R(C)=R(PC)=R
EnO
7
同济大学
4.5.
(i)AB=O(ii)AB=O
A
= B=O.
(i)′AX=AY
A
= X=Y.= X=Y.
EnO
B
= A=O.
(ii)′XB=YB
B
(i)Am×n
= AP,
. B=
BO .
O=PAB=
EnO
4.6.R(A )=
n,R(A)=n; 0,1,
R(A)=n 1;R(A)≤n 2.
= A
A =|A|n 2A.
(i)R(A)=n= A
= R(A )=n.(n 1
(ii)R(A)≤n 2= (iii)
)
= A =0.…… 此处隐藏：2449字，全部文档内容请下载后查看。喜欢就下载吧 ……
线性代数笔记第三章.doc
将本文的Word文档下载到电脑
