:: Associated Matrix of Linear Map
:: by Robert Milewski
::
:: Received June 30, 1995
:: Copyright (c) 1995-2021 Association of Mizar Users


definition
let D be non empty set ;
let k be Nat;
let M be Matrix of D;
:: original: DelLine
redefine func Del (M,k) -> Matrix of D;
coherence
DelLine (,M) is Matrix of D
proof end;
end;

theorem :: MATRLIN:1
canceled;

Th1: for n being Nat
for M being FinSequence st len M = n + 1 holds
len (Del (M,(n + 1))) = n
by PRE_POLY:12;

theorem Th2: :: MATRLIN:2
for m, n being Nat
for D being non empty set
for M being Matrix of n + 1,m,D
for M1 being Matrix of D holds
( ( n > 0 implies width M = width (Del (M,(n + 1))) ) & ( M1 = <*(M . (n + 1))*> implies width M = width M1 ) )
proof end;

theorem Th3: :: MATRLIN:3
for m, n being Nat
for D being non empty set
for M being Matrix of n + 1,m,D holds Del (M,(n + 1)) is Matrix of n,m,D
proof end;

theorem :: MATRLIN:4
canceled;

Th4: for M being FinSequence st M <> {} holds
M = (Del (M,(len M))) ^ <*(M . (len M))*>
by PRE_POLY:13;

definition
let D be non empty set ;
let P be FinSequence of D;
:: original: <*
redefine func <*P*> -> Matrix of 1, len P,D;
coherence
<*P*> is Matrix of 1, len P,D by MATRIX_0:11;
end;

theorem Th5: :: MATRLIN:5
for K being Field
for V being VectSp of K
for KL1, KL2 being Linear_Combination of V
for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Sum KL1 = Sum KL2 holds
KL1 = KL2
proof end;

theorem Th6: :: MATRLIN:6
for K being Field
for V being VectSp of K
for KL1, KL2, KL3 being Linear_Combination of V
for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & Carrier KL3 c= X & Sum KL1 = (Sum KL2) + (Sum KL3) holds
KL1 = KL2 + KL3
proof end;

theorem Th7: :: MATRLIN:7
for K being Field
for V being VectSp of K
for a being Element of K
for KL1, KL2 being Linear_Combination of V
for X being Subset of V st X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X & a <> 0. K & Sum KL1 = a * (Sum KL2) holds
KL1 = a * KL2
proof end;

theorem Th8: :: MATRLIN:8
for K being Field
for V being VectSp of K
for W being Element of V
for b2 being Basis of V ex KL being Linear_Combination of V st
( W = Sum KL & Carrier KL c= b2 )
proof end;

definition
let K be Field;
let V be VectSp of K;
attr V is finite-dimensional means :Def1: :: MATRLIN:def 1
 ex A being finite Subset of V st A is Basis of V;
end;

:: deftheorem Def1 defines finite-dimensional MATRLIN:def 1 :
for K being Field
for V being VectSp of K holds
( V is finite-dimensional iff ex A being finite Subset of V st A is Basis of V );

registration
let K be Field;
cluster non empty right_complementable strict vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed finite-dimensional for ModuleStr over K;
existence
ex b1 being VectSp of K st
( b1 is strict & b1 is finite-dimensional )
proof end;
end;

definition
let K be Field;
let V be finite-dimensional VectSp of K;
mode OrdBasis of V -> FinSequence of V means :Def2: :: MATRLIN:def 2
( it is one-to-one & rng it is Basis of V );
existence
ex b1 being FinSequence of V st
( b1 is one-to-one & rng b1 is Basis of V )
proof end;
end;

:: deftheorem Def2 defines OrdBasis MATRLIN:def 2 :
for K being Field
for V being finite-dimensional VectSp of K
for b3 being FinSequence of V holds
( b3 is OrdBasis of V iff ( b3 is one-to-one & rng b3 is Basis of V ) );

definition
let K be non empty doubleLoopStr ;
let V1, V2 be non empty ModuleStr over K;
let f1, f2 be Function of V1,V2;
func f1 + f2 -> Function of V1,V2 means :Def3: :: MATRLIN:def 3
 for v being Element of V1 holds it . v = (f1 . v) + (f2 . v);
existence
ex b1 being Function of V1,V2 st
for v being Element of V1 holds b1 . v = (f1 . v) + (f2 . v)
proof end;
uniqueness
for b1, b2 being Function of V1,V2 st ( for v being Element of V1 holds b1 . v = (f1 . v) + (f2 . v) ) & ( for v being Element of V1 holds b2 . v = (f1 . v) + (f2 . v) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def3 defines + MATRLIN:def 3 :
for K being non empty doubleLoopStr
for V1, V2 being non empty ModuleStr over K
for f1, f2, b6 being Function of V1,V2 holds
( b6 = f1 + f2 iff for v being Element of V1 holds b6 . v = (f1 . v) + (f2 . v) );

definition
let K be non empty doubleLoopStr ;
let V1, V2 be non empty ModuleStr over K;
let f be Function of V1,V2;
let a be Element of K;
func a * f -> Function of V1,V2 means :Def4: :: MATRLIN:def 4
 for v being Element of V1 holds it . v = a * (f . v);
existence
ex b1 being Function of V1,V2 st
for v being Element of V1 holds b1 . v = a * (f . v)
proof end;
uniqueness
for b1, b2 being Function of V1,V2 st ( for v being Element of V1 holds b1 . v = a * (f . v) ) & ( for v being Element of V1 holds b2 . v = a * (f . v) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def4 defines * MATRLIN:def 4 :
for K being non empty doubleLoopStr
for V1, V2 being non empty ModuleStr over K
for f being Function of V1,V2
for a being Element of K
for b6 being Function of V1,V2 holds
( b6 = a * f iff for v being Element of V1 holds b6 . v = a * (f . v) );

theorem Th9: :: MATRLIN:9
for K being Field
for V1 being finite-dimensional VectSp of K
for a being Element of V1
for F being FinSequence of V1
for G being FinSequence of K st len F = len G & ( for k being Nat
for v being Element of K st k in dom F & v = G . k holds
F . k = v * a ) holds
Sum F = (Sum G) * a
proof end;

theorem Th10: :: MATRLIN:10
for K being Field
for V1 being finite-dimensional VectSp of K
for a being Element of V1
for F being FinSequence of K
for G being FinSequence of V1 st len F = len G & ( for k being Nat st k in dom F holds
G . k = (F /. k) * a ) holds
Sum G = (Sum F) * a
proof end;

theorem Th11: :: MATRLIN:11
for V1 being non empty right_complementable add-associative right_zeroed addLoopStr
for F being FinSequence of V1 st ( for k being Nat st k in dom F holds
F /. k = 0. V1 ) holds
Sum F = 0. V1
proof end;

definition
let K be Field;
let V1 be finite-dimensional VectSp of K;
let p1 be FinSequence of K;
let p2 be FinSequence of V1;
func lmlt (p1,p2) -> FinSequence of V1 equals :: MATRLIN:def 5
 the lmult of V1 .: (p1,p2);
coherence
the lmult of V1 .: (p1,p2) is FinSequence of V1 ;
end;

:: deftheorem defines lmlt MATRLIN:def 5 :
for K being Field
for V1 being finite-dimensional VectSp of K
for p1 being FinSequence of K
for p2 being FinSequence of V1 holds lmlt (p1,p2) = the lmult of V1 .: (p1,p2);

theorem Th12: :: MATRLIN:12
for K being Field
for V1 being finite-dimensional VectSp of K
for p2 being FinSequence of V1
for p1 being FinSequence of K st dom p1 = dom p2 holds
dom (lmlt (p1,p2)) = dom p1
proof end;

definition
let V1 be non empty addLoopStr ;
let M be FinSequence of the carrier of V1 * ;
func Sum M -> FinSequence of V1 means :Def6: :: MATRLIN:def 6
( len it = len M & ( for k being Nat st k in dom it holds
it /. k = Sum (M /. k) ) );
existence
ex b1 being FinSequence of V1 st
( len b1 = len M & ( for k being Nat st k in dom b1 holds
b1 /. k = Sum (M /. k) ) )
proof end;
uniqueness
for b1, b2 being FinSequence of V1 st len b1 = len M & ( for k being Nat st k in dom b1 holds
b1 /. k = Sum (M /. k) ) & len b2 = len M & ( for k being Nat st k in dom b2 holds
b2 /. k = Sum (M /. k) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def6 defines Sum MATRLIN:def 6 :
for V1 being non empty addLoopStr
for M being FinSequence of the carrier of V1 *
for b3 being FinSequence of V1 holds
( b3 = Sum M iff ( len b3 = len M & ( for k being Nat st k in dom b3 holds
b3 /. k = Sum (M /. k) ) ) );

theorem Th13: :: MATRLIN:13
for K being Field
for V1 being finite-dimensional VectSp of K
for M being Matrix of the carrier of V1 st len M = 0 holds
Sum (Sum M) = 0. V1
proof end;

theorem Th14: :: MATRLIN:14
for m being Nat
for K being Field
for V1 being finite-dimensional VectSp of K
for M being Matrix of m + 1, 0 , the carrier of V1 holds Sum (Sum M) = 0. V1
proof end;

theorem Th15: :: MATRLIN:15
for D being non empty set
for x being Element of D holds <*<*x*>*> = <*<*x*>*> @
proof end;

theorem Th16: :: MATRLIN:16
for K being Field
for V1, V2 being VectSp of K
for f being Function of V1,V2
for p being FinSequence of V1 st f is additive & f is homogeneous holds
f . (Sum p) = Sum (f * p)
proof end;

theorem Th17: :: MATRLIN:17
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f being Function of V1,V2
for a being FinSequence of K
for p being FinSequence of V1 st len p = len a & f is additive & f is homogeneous holds
f * (lmlt (a,p)) = lmlt (a,(f * p))
proof end;

theorem Th18: :: MATRLIN:18
for K being Field
for V2, V3 being finite-dimensional VectSp of K
for g being Function of V2,V3
for b2 being OrdBasis of V2
for a being FinSequence of K st len a = len b2 & g is additive & g is homogeneous holds
g . (Sum (lmlt (a,b2))) = Sum (lmlt (a,(g * b2)))
proof end;

theorem Th19: :: MATRLIN:19
for K being Field
for V1 being finite-dimensional VectSp of K
for F, F1 being FinSequence of V1
for KL being Linear_Combination of V1
for p being Permutation of (dom F) st F1 = F * p holds
KL (#) F1 = (KL (#) F) * p
proof end;

theorem Th20: :: MATRLIN:20
for K being Field
for V1 being finite-dimensional VectSp of K
for F being FinSequence of V1
for KL being Linear_Combination of V1 st F is one-to-one & Carrier KL c= rng F holds
Sum (KL (#) F) = Sum KL
proof end;

theorem Th21: :: MATRLIN:21
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f1, f2 being Function of V1,V2
for A being set
for p being FinSequence of V1 st rng p c= A & f1 is additive & f1 is homogeneous & f2 is additive & f2 is homogeneous & ( for v being Element of V1 st v in A holds
f1 . v = f2 . v ) holds
f1 . (Sum p) = f2 . (Sum p)
proof end;

theorem Th22: :: MATRLIN:22
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f1, f2 being Function of V1,V2 st f1 is additive & f1 is homogeneous & f2 is additive & f2 is homogeneous holds
for b1 being OrdBasis of V1 st len b1 > 0 & f1 * b1 = f2 * b1 holds
f1 = f2
proof end;

registration
let D be non empty set ;
cluster tabular -> FinSequence-yielding for FinSequence of D * ;
coherence
for b1 being Matrix of D holds b1 is FinSequence-yielding ;
end;

definition
let D be non empty set ;
let F, G be Matrix of D;
:: original: ^^
redefine func F ^^ G -> Matrix of D;
coherence
F ^^ G is Matrix of D
proof end;
end;

definition
let D be non empty set ;
let n, m, k be Nat;
let M1 be Matrix of n,k,D;
let M2 be Matrix of m,k,D;
:: original: ^
redefine func M1 ^ M2 -> Matrix of n + m,k,D;
coherence
M1 ^ M2 is Matrix of n + m,k,D
proof end;
end;

theorem :: MATRLIN:23
for k, i, m, n being Nat
for D being non empty set
for M1 being Matrix of n,k,D
for M2 being Matrix of m,k,D st i in dom M1 holds
Line ((M1 ^ M2),i) = Line (M1,i)
proof end;

theorem Th24: :: MATRLIN:24
for k, m, n being Nat
for D being non empty set
for M1 being Matrix of n,k,D
for M2 being Matrix of m,k,D st width M1 = width M2 holds
width (M1 ^ M2) = width M1
proof end;

theorem :: MATRLIN:25
for k, t, i, m, n being Nat
for D being non empty set
for M1 being Matrix of t,k,D
for M2 being Matrix of m,k,D st n in dom M2 & i = (len M1) + n holds
Line ((M1 ^ M2),i) = Line (M2,n)
proof end;

theorem Th26: :: MATRLIN:26
for k, m, n being Nat
for D being non empty set
for M1 being Matrix of n,k,D
for M2 being Matrix of m,k,D st width M1 = width M2 holds
for i being Nat st i in Seg (width M1) holds
Col ((M1 ^ M2),i) = (Col (M1,i)) ^ (Col (M2,i))
proof end;

theorem Th27: :: MATRLIN:27
for k, m, n being Nat
for K being Field
for V being VectSp of K
for M1 being Matrix of n,k, the carrier of V
for M2 being Matrix of m,k, the carrier of V holds Sum (M1 ^ M2) = (Sum M1) ^ (Sum M2)
proof end;

theorem Th28: :: MATRLIN:28
for k, m, n being Nat
for D being non empty set
for M1 being Matrix of n,k,D
for M2 being Matrix of m,k,D st width M1 = width M2 holds
(M1 ^ M2) @ = (M1 @) ^^ (M2 @)
proof end;

theorem Th29: :: MATRLIN:29
for K being Field
for V1 being finite-dimensional VectSp of K
for M1, M2 being Matrix of the carrier of V1 holds (Sum M1) + (Sum M2) = Sum (M1 ^^ M2)
proof end;

theorem Th30: :: MATRLIN:30
for K being Field
for V1 being finite-dimensional VectSp of K
for P1, P2 being FinSequence of V1 st len P1 = len P2 holds
Sum (P1 + P2) = (Sum P1) + (Sum P2)
proof end;

theorem Th31: :: MATRLIN:31
for K being Field
for V1 being finite-dimensional VectSp of K
for M1, M2 being Matrix of the carrier of V1 st len M1 = len M2 holds
(Sum (Sum M1)) + (Sum (Sum M2)) = Sum (Sum (M1 ^^ M2))
proof end;

theorem Th32: :: MATRLIN:32
for K being Field
for V1 being finite-dimensional VectSp of K
for M being Matrix of the carrier of V1 holds Sum (Sum M) = Sum (Sum (M @))
proof end;

theorem Th33: :: MATRLIN:33
for m, n being Nat
for K being Field
for V1 being finite-dimensional VectSp of K
for M being Matrix of n,m, the carrier of K st n > 0 & m > 0 holds
for p, d being FinSequence of K st len p = n & len d = m & ( for j being Nat st j in dom d holds
d /. j = Sum (mlt (p,(Col (M,j)))) ) holds
for b, c being FinSequence of V1 st len b = m & len c = n & ( for i being Nat st i in dom c holds
c /. i = Sum (lmlt ((Line (M,i)),b)) ) holds
Sum (lmlt (p,c)) = Sum (lmlt (d,b))
proof end;

definition
let K be Field;
let V be finite-dimensional VectSp of K;
let b1 be OrdBasis of V;
let W be Element of V;
func W |-- b1 -> FinSequence of K means :Def7: :: MATRLIN:def 7
( len it = len b1 & ex KL being Linear_Combination of V st
( W = Sum KL & Carrier KL c= rng b1 & ( for k being Nat st 1 <= k & k <= len it holds
it /. k = KL . (b1 /. k) ) ) );
existence
ex b1 being FinSequence of K st
( len b1 = len b1 & ex KL being Linear_Combination of V st
( W = Sum KL & Carrier KL c= rng b1 & ( for k being Nat st 1 <= k & k <= len b1 holds
b1 /. k = KL . (b1 /. k) ) ) )
proof end;
uniqueness
for b1, b2 being FinSequence of K st len b1 = len b1 & ex KL being Linear_Combination of V st
( W = Sum KL & Carrier KL c= rng b1 & ( for k being Nat st 1 <= k & k <= len b1 holds
b1 /. k = KL . (b1 /. k) ) ) & len b2 = len b1 & ex KL being Linear_Combination of V st
( W = Sum KL & Carrier KL c= rng b1 & ( for k being Nat st 1 <= k & k <= len b2 holds
b2 /. k = KL . (b1 /. k) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines |-- MATRLIN:def 7 :
for K being Field
for V being finite-dimensional VectSp of K
for b1 being OrdBasis of V
for W being Element of V
for b5 being FinSequence of K holds
( b5 = W |-- b1 iff ( len b5 = len b1 & ex KL being Linear_Combination of V st
( W = Sum KL & Carrier KL c= rng b1 & ( for k being Nat st 1 <= k & k <= len b5 holds
b5 /. k = KL . (b1 /. k) ) ) ) );

Lm1: for K being Field
for V being finite-dimensional VectSp of K
for b being OrdBasis of V
for W being Element of V holds dom (W |-- b) = dom b
proof end;

theorem Th34: :: MATRLIN:34
for K being Field
for V2 being finite-dimensional VectSp of K
for b2 being OrdBasis of V2
for v1, v2 being Vector of V2 st v1 |-- b2 = v2 |-- b2 holds
v1 = v2
proof end;

theorem Th35: :: MATRLIN:35
for K being Field
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for v being Element of V1 holds v = Sum (lmlt ((v |-- b1),b1))
proof end;

theorem Th36: :: MATRLIN:36
for K being Field
for V1 being finite-dimensional VectSp of K
for b1 being OrdBasis of V1
for d being FinSequence of K st len d = len b1 holds
d = (Sum (lmlt (d,b1))) |-- b1
proof end;

Lm2: for p being FinSequence
for k being set st k in dom p holds
len p > 0
proof end;

theorem Th37: :: MATRLIN:37
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for a, d being FinSequence of K st len a = len b1 holds
for j being Nat st j in dom b2 & len d = len b1 & ( for k being Nat st k in dom b1 holds
d . k = ((f . (b1 /. k)) |-- b2) /. j ) & len b1 > 0 holds
((Sum (lmlt (a,(f * b1)))) |-- b2) /. j = Sum (mlt (a,d))
proof end;

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;
func AutMt (f,b1,b2) -> Matrix of K means :Def8: :: MATRLIN:def 8
( len it = len b1 & ( for k being Nat st k in dom b1 holds
it /. k = (f . (b1 /. k)) |-- b2 ) );
existence
ex b1 being Matrix of K st
( len b1 = len b1 & ( for k being Nat st k in dom b1 holds
b1 /. k = (f . (b1 /. k)) |-- b2 ) )
proof end;
uniqueness
for b1, b2 being Matrix of K st len b1 = len b1 & ( for k being Nat st k in dom b1 holds
b1 /. k = (f . (b1 /. k)) |-- b2 ) & len b2 = len b1 & ( for k being Nat st k in dom b1 holds
b2 /. k = (f . (b1 /. k)) |-- b2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines AutMt MATRLIN:def 8 :
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f being Function of V1,V2
for b1 being FinSequence of V1
for b2 being OrdBasis of V2
for b7 being Matrix of K holds
( b7 = AutMt (f,b1,b2) iff ( len b7 = len b1 & ( for k being Nat st k in dom b1 holds
b7 /. k = (f . (b1 /. k)) |-- b2 ) ) );

Lm3: for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 holds dom (AutMt (f,b1,b2)) = dom b1
proof end;

theorem Th38: :: MATRLIN:38
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 st len b1 = 0 holds
AutMt (f,b1,b2) = {}
proof end;

theorem Th39: :: MATRLIN:39
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 st len b1 > 0 holds
width (AutMt (f,b1,b2)) = len b2
proof end;

theorem :: MATRLIN:40
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f1, f2 being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 st f1 is additive & f1 is homogeneous & f2 is additive & f2 is homogeneous & AutMt (f1,b1,b2) = AutMt (f2,b1,b2) & len b1 > 0 holds
f1 = f2
proof end;

theorem :: MATRLIN:41
for K being Field
for V1, V2, V3 being finite-dimensional VectSp of K
for f being Function of V1,V2
for g being Function of V2,V3
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2
for b3 being OrdBasis of V3 st g is additive & g is homogeneous & len b1 > 0 & len b2 > 0 holds
AutMt ((g * f),b1,b3) = (AutMt (f,b1,b2)) * (AutMt (g,b2,b3))
proof end;

theorem :: MATRLIN:42
for K being Field
for V1, V2 being finite-dimensional VectSp of K
for f1, f2 being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 holds AutMt ((f1 + f2),b1,b2) = (AutMt (f1,b1,b2)) + (AutMt (f2,b1,b2))
proof end;

theorem :: MATRLIN:43
for K being Field
for a being Element of K
for V1, V2 being finite-dimensional VectSp of K
for f being Function of V1,V2
for b1 being OrdBasis of V1
for b2 being OrdBasis of V2 st a <> 0. K holds
AutMt ((a * f),b1,b2) = a * (AutMt (f,b1,b2))
proof end;