:: Linear Map of Matrices
:: by Karol P\c{a}k
::
:: Received May 13, 2008
:: Copyright (c) 2008 Association of Mizar Users
theorem Th1: :: MATRLIN2:1
theorem Th2: :: MATRLIN2:2
theorem Th3: :: MATRLIN2:3
theorem :: MATRLIN2:4
theorem :: MATRLIN2:5
theorem :: MATRLIN2:6
Lm1:
for K being Field
for V1 being finite-dimensional VectSp of K
for R being FinSequence of V1
for p being FinSequence of K holds dom (lmlt p,R) = (dom p) /\ (dom R)
Lm2:
for K being Field
for p1, p2 being FinSequence of K holds dom (p1 + p2) = (dom p1) /\ (dom p2)
Lm3:
for K being Field
for V1 being finite-dimensional VectSp of K
for R1, R2 being FinSequence of V1 holds dom (R1 + R2) = (dom R1) /\ (dom R2)
theorem Th7: :: MATRLIN2:7
theorem :: MATRLIN2:8
theorem Th9: :: MATRLIN2:9
theorem :: MATRLIN2:10
theorem :: MATRLIN2:11
theorem :: MATRLIN2:12
theorem :: MATRLIN2:13
theorem :: MATRLIN2:14
theorem :: MATRLIN2:15
theorem :: MATRLIN2:16
theorem Th17: :: MATRLIN2:17
theorem Th18: :: MATRLIN2:18
theorem Th19: :: MATRLIN2:19
theorem Th20: :: MATRLIN2:20
theorem Th21: :: MATRLIN2:21
Lm4:
for K being Field
for V being VectSp of K
for W1 being Subspace of V
for L1 being Linear_Combination of W1 ex K1 being Linear_Combination of V st
( Carrier K1 = Carrier L1 & Sum K1 = Sum L1 & K1 | the carrier of W1 = L1 )
theorem :: MATRLIN2:22
theorem :: MATRLIN2:23
theorem Th24: :: MATRLIN2:24
theorem Th25: :: MATRLIN2:25
theorem Th26: :: MATRLIN2:26
definition
let K be
Field;
let V1,
V2 be
finite-dimensional VectSp of
K;
let f be
Function of
V1,
V2;
let B1 be
FinSequence of
V1;
let b2 be
OrdBasis of
V2;
:: original: AutMtredefine func AutMt f,
B1,
b2 -> Matrix of
len B1,
len b2,
K;
coherence
AutMt f,B1,b2 is Matrix of len B1, len b2,K
end;
:: deftheorem defines | MATRLIN2:def 1 :
theorem :: MATRLIN2:27
for
K being
Field for
V1,
V2 being
finite-dimensional VectSp of
K for
b1 being
OrdBasis of
V1 for
b2 being
OrdBasis of
V2 for
f being
linear-transformation of
V1,
V2 for
W1,
W2 being
Subspace of
V1 for
U1,
U2 being
Subspace of
V2 st (
dim W1 = 0 implies
dim U1 = 0 ) & (
dim W2 = 0 implies
dim U2 = 0 ) &
V2 is_the_direct_sum_of U1,
U2 holds
for
fW1 being
linear-transformation of
W1,
U1 for
fW2 being
linear-transformation of
W2,
U2 st
fW1 = f | W1 &
fW2 = f | W2 holds
for
w1 being
OrdBasis of
W1 for
w2 being
OrdBasis of
W2 for
u1 being
OrdBasis of
U1 for
u2 being
OrdBasis of
U2 st
w1 ^ w2 = b1 &
u1 ^ u2 = b2 holds
AutMt f,
b1,
b2 = block_diagonal <*(AutMt fW1,w1,u1),(AutMt fW2,w2,u2)*>,
(0. K)
definition
let K be
Field;
let V1,
V2 be
finite-dimensional VectSp of
K;
let f be
Function of
V1,
V2;
let B1 be
FinSequence of
V1;
let b2 be
OrdBasis of
V2;
assume A1:
len B1 = len b2
;
func AutEqMt f,
B1,
b2 -> Matrix of
len B1,
len B1,
K equals :
Def2:
:: MATRLIN2:def 2
AutMt f,
B1,
b2;
coherence
AutMt f,B1,b2 is Matrix of len B1, len B1,K
by A1;
end;
:: deftheorem Def2 defines AutEqMt MATRLIN2:def 2 :
theorem Th28: :: MATRLIN2:28
theorem :: MATRLIN2:29
theorem Th30: :: MATRLIN2:30
theorem Th31: :: MATRLIN2:31
definition
let K be
Field;
let V1,
V2 be
finite-dimensional VectSp of
K;
let b1 be
OrdBasis of
V1;
let B2 be
FinSequence of
V2;
let M be
Matrix of
len b1,
len B2,
K;
func Mx2Tran M,
b1,
B2 -> Function of
V1,
V2 means :
Def3:
:: MATRLIN2:def 3
for
v being
Vector of
V1 holds
it . v = Sum (lmlt (Line ((LineVec2Mx (v |-- b1)) * M),1),B2);
existence
ex b1 being Function of V1,V2 st
for v being Vector of V1 holds b1 . v = Sum (lmlt (Line ((LineVec2Mx (v |-- b1)) * M),1),B2)
uniqueness
for b1, b2 being Function of V1,V2 st ( for v being Vector of V1 holds b1 . v = Sum (lmlt (Line ((LineVec2Mx (v |-- b1)) * M),1),B2) ) & ( for v being Vector of V1 holds b2 . v = Sum (lmlt (Line ((LineVec2Mx (v |-- b1)) * M),1),B2) ) holds
b1 = b2
end;
:: deftheorem Def3 defines Mx2Tran MATRLIN2:def 3 :
theorem Th32: :: MATRLIN2:32
theorem Th33: :: MATRLIN2:33
Lm5:
for K being Field
for A, B being Matrix of K st len A = len B & width A = width B holds
for i being Nat st 1 <= i & i <= len A holds
Line (A + B),i = (Line A,i) + (Line B,i)
theorem Th34: :: MATRLIN2:34
theorem Th35: :: MATRLIN2:35
theorem Th36: :: MATRLIN2:36
theorem :: MATRLIN2:37
for
K being
Field for
V1,
V2 being
finite-dimensional VectSp of
K for
b1 being
OrdBasis of
V1 for
B2 being
FinSequence of
V2 for
A,
B being
Matrix of
len b1,
len B2,
K holds
Mx2Tran (A + B),
b1,
B2 = (Mx2Tran A,b1,B2) + (Mx2Tran B,b1,B2)
theorem :: MATRLIN2:38
theorem :: MATRLIN2:39
theorem :: MATRLIN2:40
for
K being
Field for
V1,
V2,
V3 being
finite-dimensional VectSp of
K for
b1 being
OrdBasis of
V1 for
b2 being
OrdBasis of
V2 for
B3 being
FinSequence of
V3 for
A being
Matrix of
len b1,
len b2,
K for
B being
Matrix of
len b2,
len B3,
K st
width A = len B holds
for
AB being
Matrix of
len b1,
len B3,
K st
AB = A * B holds
Mx2Tran AB,
b1,
B3 = (Mx2Tran B,b2,B3) * (Mx2Tran A,b1,b2)
theorem Th41: :: MATRLIN2:41
theorem Th42: :: MATRLIN2:42
theorem Th43: :: MATRLIN2:43
theorem Th44: :: MATRLIN2:44
Lm6:
for n being Nat
for K being Field holds the_rank_of (1. K,n) = n
theorem Th45: :: MATRLIN2:45
theorem Th46: :: MATRLIN2:46
theorem Th47: :: MATRLIN2:47
Lm7:
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for A being Matrix of len b1, len b2,K st len b1 > 0 & len b2 > 0 holds
nullity (Mx2Tran A,b1,b2) = (len b1) - (the_rank_of A)
theorem Th48: :: MATRLIN2:48
theorem :: MATRLIN2:49
theorem :: MATRLIN2:50
Lm8:
for K being Field
for V1, V2, V3 being finite-dimensional VectSp of K
for f being linear-transformation of V1,V2
for g being Function of V2,V3 holds g * f = (g | (im f)) * f
theorem :: MATRLIN2:51