:: 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;