RLVECT_2: Linear Combinations in Real Linear Space

:: Linear Combinations in Real Linear Space
:: by Wojciech A. Trybulec
::
:: Received April 8, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let S be 1-sorted ;
let x be set ;
assume A1: x in S ;

func vector (S,x) -> Element of S equals :Def1: :: RLVECT_2:def 1
x;

coherence by A1;

end;

:: deftheorem Def1 defines vector RLVECT_2:def 1 :

theorem :: RLVECT_2:1

for S being non empty 1-sorted
for v being Element of S holds vector (S,v) = v by Def1, RLVECT_1:1;

theorem Th2: :: RLVECT_2:2

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G, H being FinSequence of the carrier of V st len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) + (G /. k) ) holds
Sum H = (Sum F) + (Sum G)

proof end;

theorem :: RLVECT_2:3

for V being RealLinearSpace
for a being Real
for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)

proof end;

theorem Th4: :: RLVECT_2:4

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V st len F = len G & ( for k being Nat st k in dom F holds
G . k = - (F /. k) ) holds
Sum G = - (Sum F)

proof end;

theorem :: RLVECT_2:5

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G, H being FinSequence of the carrier of V st len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)

proof end;

theorem Th6: :: RLVECT_2:6

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V
for f being Permutation of (dom F) st len F = len G & ( for i being Nat st i in dom G holds
G . i = F . (f . i) ) holds
Sum F = Sum G

proof end;

theorem :: RLVECT_2:7

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for F, G being FinSequence of the carrier of V
for f being Permutation of (dom F) st G = F * f holds
Sum F = Sum G

proof end;

definition

let V be non empty addLoopStr ;
let T be finite Subset of V;
assume A1: ( V is Abelian & V is add-associative & V is right_zeroed ) ;

func Sum T -> Element of V means :Def2: :: RLVECT_2:def 2
ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & it = Sum F );

proof end;

uniqueness by A1, RLVECT_1:42;

end;

:: deftheorem Def2 defines Sum RLVECT_2:def 2 :

theorem Th8: :: RLVECT_2:8

for V being non empty Abelian add-associative right_zeroed addLoopStr holds Sum ({} V) = 0. V

proof end;

theorem :: RLVECT_2:9

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v being Element of V holds Sum {v} = v

proof end;

theorem :: RLVECT_2:10

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v1, v2 being Element of V st v1 <> v2 holds
Sum {v1,v2} = v1 + v2

proof end;

theorem :: RLVECT_2:11

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for v1, v2, v3 being Element of V st v1 <> v2 & v2 <> v3 & v1 <> v3 holds
Sum {v1,v2,v3} = (v1 + v2) + v3

proof end;

theorem Th12: :: RLVECT_2:12

for V being non empty Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V st T misses S holds
Sum (T \/ S) = (Sum T) + (Sum S)

proof end;

theorem Th13: :: RLVECT_2:13

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S))

proof end;

theorem :: RLVECT_2:14

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T /\ S) = ((Sum T) + (Sum S)) - (Sum (T \/ S))

proof end;

theorem Th15: :: RLVECT_2:15

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \ S) = (Sum (T \/ S)) - (Sum S)

proof end;

theorem Th16: :: RLVECT_2:16

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \ S) = (Sum T) - (Sum (T /\ S))

proof end;

theorem :: RLVECT_2:17

for V being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \/ S)) - (Sum (T /\ S))

proof end;

theorem :: RLVECT_2:18

for V being non empty Abelian add-associative right_zeroed addLoopStr
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \ S)) + (Sum (S \ T)) by Th12, XBOOLE_1:82;

reconsider zz = 0 as Element of REAL by XREAL_0:def 1;

definition

let V be non empty ZeroStr ;

mode Linear_Combination of V -> Element of Funcs ( the carrier of V,REAL) means :Def3: :: RLVECT_2:def 3
ex T being finite Subset of V st
for v being Element of V st not v in T holds
it . v = 0 ;

proof end;

end;

:: deftheorem Def3 defines Linear_Combination RLVECT_2:def 3 :

notation

let V be non empty addLoopStr ;
let L be Element of Funcs ( the carrier of V,REAL);
synonym Carrier L for support V;

end;

Lm1: now :: thesis:

let V be non empty addLoopStr ; :: thesis:
let L be Element of Funcs ( the carrier of V,REAL); :: thesis:
A1: support L c= dom L by PRE_POLY:37;
thus Carrier c= the carrier of V :: thesis:

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in support L ; :: thesis:
then x in dom L by A1;
hence x in the carrier of V ; :: thesis:

end;

end;

definition

let V be non empty addLoopStr ;
let L be Element of Funcs ( the carrier of V,REAL);

:: original: Carrier
redefine func Carrier L -> Subset of V equals :: RLVECT_2:def 4
{ v where v is Element of V : L . v <> 0 } ;

coherence by Lm1;
compatibility

proof end;

end;

:: deftheorem defines Carrier RLVECT_2:def 4 :

registration

let V be non empty addLoopStr ;
let L be Linear_Combination of V;

cluster Carrier -> finite ;

proof end;

end;

theorem :: RLVECT_2:19

for V being non empty addLoopStr
for L being Linear_Combination of V
for v being Element of V holds
( L . v = 0 iff not v in Carrier L )

proof end;

definition

let V be non empty addLoopStr ;

func ZeroLC V -> Linear_Combination of V means :Def5: :: RLVECT_2:def 5
Carrier it = {} ;

proof end;

proof end;

end;

:: deftheorem Def5 defines ZeroLC RLVECT_2:def 5 :

theorem Th20: :: RLVECT_2:20

for V being non empty addLoopStr
for v being Element of V holds (ZeroLC V) . v = 0

proof end;

definition

let V be non empty addLoopStr ;
let A be Subset of V;

mode Linear_Combination of A -> Linear_Combination of V means :Def6: :: RLVECT_2:def 6
Carrier it c= A;

proof end;

end;

:: deftheorem Def6 defines Linear_Combination RLVECT_2:def 6 :

theorem :: RLVECT_2:21

for V being RealLinearSpace
for A, B being Subset of V
for l being Linear_Combination of A st A c= B holds
l is Linear_Combination of B

proof end;

theorem Th22: :: RLVECT_2:22

for V being RealLinearSpace
for A being Subset of V holds ZeroLC V is Linear_Combination of A

proof end;

theorem Th23: :: RLVECT_2:23

for V being RealLinearSpace
for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V

proof end;

definition

let V be RealLinearSpace;
let F be FinSequence of V;
let f be Function of the carrier of V,REAL;

func f (#) F -> FinSequence of the carrier of V means :Def7: :: RLVECT_2:def 7
( len it = len F & ( for i being Nat st i in dom it holds
it . i = (f . (F /. i)) * (F /. i) ) );

proof end;

proof end;

end;

:: deftheorem Def7 defines (#) RLVECT_2:def 7 :

theorem Th24: :: RLVECT_2:24

for i being Nat
for V being RealLinearSpace
for v being VECTOR of V
for F being FinSequence of V
for f being Function of the carrier of V,REAL st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v

proof end;

theorem :: RLVECT_2:25

for V being RealLinearSpace
for f being Function of the carrier of V,REAL holds f (#) (<*> the carrier of V) = <*> the carrier of V

proof end;

theorem Th26: :: RLVECT_2:26

for V being RealLinearSpace
for v being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v*> = <*((f . v) * v)*>

proof end;

theorem Th27: :: RLVECT_2:27

for V being RealLinearSpace
for v1, v2 being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>

proof end;

theorem :: RLVECT_2:28

for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>

proof end;

definition

let V be RealLinearSpace;
let L be Linear_Combination of V;

func Sum L -> Element of V means :Def8: :: RLVECT_2:def 8
ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & it = Sum (L (#) F) );

proof end;

proof end;

end;

:: deftheorem Def8 defines Sum RLVECT_2:def 8 :

Lm2: for V being RealLinearSpace holds Sum (ZeroLC V) = 0. V

proof end;

theorem :: RLVECT_2:29

for V being RealLinearSpace
for A being Subset of V holds
( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )

proof end;

theorem :: RLVECT_2:30

for V being RealLinearSpace holds Sum (ZeroLC V) = 0. V by Lm2;

theorem :: RLVECT_2:31

for V being RealLinearSpace
for l being Linear_Combination of {} the carrier of V holds Sum l = 0. V

proof end;

theorem Th32: :: RLVECT_2:32

for V being RealLinearSpace
for v being VECTOR of V
for l being Linear_Combination of {v} holds Sum l = (l . v) * v

proof end;

theorem Th33: :: RLVECT_2:33

for V being RealLinearSpace
for v1, v2 being VECTOR of V st v1 <> v2 holds
for l being Linear_Combination of {v1,v2} holds Sum l = ((l . v1) * v1) + ((l . v2) * v2)

proof end;

theorem :: RLVECT_2:34

for V being RealLinearSpace
for L being Linear_Combination of V st Carrier L = {} holds
Sum L = 0. V

proof end;

theorem :: RLVECT_2:35

for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V st Carrier L = {v} holds
Sum L = (L . v) * v

proof end;

theorem :: RLVECT_2:36

for V being RealLinearSpace
for v1, v2 being VECTOR of V
for L being Linear_Combination of V st Carrier L = {v1,v2} & v1 <> v2 holds
Sum L = ((L . v1) * v1) + ((L . v2) * v2)

proof end;

definition

let V be non empty addLoopStr ;
let L1, L2 be Linear_Combination of V;

:: original: =
redefine pred L1 = L2 means :: RLVECT_2:def 9
for v being Element of V holds L1 . v = L2 . v;

compatibility ;

end;

:: deftheorem defines = RLVECT_2:def 9 :

definition

let V be non empty addLoopStr ;
let L1, L2 be Linear_Combination of V;

:: original: +
redefine func L1 + L2 -> Linear_Combination of V means :Def10: :: RLVECT_2:def 10
for v being Element of V holds it . v = (L1 . v) + (L2 . v);

proof end;

proof end;

end;

:: deftheorem Def10 defines + RLVECT_2:def 10 :

theorem Th37: :: RLVECT_2:37

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2)

proof end;

theorem Th38: :: RLVECT_2:38

for V being RealLinearSpace
for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 + L2 is Linear_Combination of A

proof end;

theorem :: RLVECT_2:39

for V being non empty addLoopStr
for L1, L2 being Linear_Combination of V holds L1 + L2 = L2 + L1 ;

theorem Th40: :: RLVECT_2:40

for V being RealLinearSpace
for L1, L2, L3 being Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3

proof end;

theorem Th41: :: RLVECT_2:41

for V being RealLinearSpace
for L being Linear_Combination of V holds
( L + (ZeroLC V) = L & (ZeroLC V) + L = L )

proof end;

definition

let V be RealLinearSpace;
let a be Real;
let L be Linear_Combination of V;

func a * L -> Linear_Combination of V means :Def11: :: RLVECT_2:def 11
for v being VECTOR of V holds it . v = a * (L . v);

proof end;

proof end;

end;

:: deftheorem Def11 defines * RLVECT_2:def 11 :

theorem Th42: :: RLVECT_2:42

for V being RealLinearSpace
for a being Real
for L being Linear_Combination of V st a <> 0 holds
Carrier (a * L) = Carrier L

proof end;

theorem Th43: :: RLVECT_2:43

for V being RealLinearSpace
for L being Linear_Combination of V holds 0 * L = ZeroLC V

proof end;

theorem Th44: :: RLVECT_2:44

for V being RealLinearSpace
for a being Real
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
a * L is Linear_Combination of A

proof end;

theorem Th45: :: RLVECT_2:45

for V being RealLinearSpace
for a, b being Real
for L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)

proof end;

theorem Th46: :: RLVECT_2:46

for V being RealLinearSpace
for a being Real
for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)

proof end;

theorem Th47: :: RLVECT_2:47

for V being RealLinearSpace
for a, b being Real
for L being Linear_Combination of V holds a * (b * L) = (a * b) * L

proof end;

theorem Th48: :: RLVECT_2:48

for V being RealLinearSpace
for L being Linear_Combination of V holds 1 * L = L

proof end;

definition

let V be RealLinearSpace;
let L be Linear_Combination of V;

func - L -> Linear_Combination of V equals :: RLVECT_2:def 12
(- 1) * L;

end;

:: deftheorem defines - RLVECT_2:def 12 :

theorem Th49: :: RLVECT_2:49

for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V holds (- L) . v = - (L . v)

proof end;

theorem :: RLVECT_2:50

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V st L1 + L2 = ZeroLC V holds
L2 = - L1

proof end;

theorem :: RLVECT_2:51

for V being RealLinearSpace
for L being Linear_Combination of V holds Carrier (- L) = Carrier L by Th42;

theorem :: RLVECT_2:52

for V being RealLinearSpace
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
- L is Linear_Combination of A by Th44;

theorem :: RLVECT_2:53

for V being RealLinearSpace
for L being Linear_Combination of V holds - (- L) = L

proof end;

definition

let V be RealLinearSpace;
let L1, L2 be Linear_Combination of V;

func L1 - L2 -> Linear_Combination of V equals :: RLVECT_2:def 13
L1 + (- L2);

end;

:: deftheorem defines - RLVECT_2:def 13 :

theorem Th54: :: RLVECT_2:54

for V being RealLinearSpace
for v being VECTOR of V
for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

proof end;

theorem :: RLVECT_2:55

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds Carrier (L1 - L2) c= (Carrier L1) \/ (Carrier L2)

proof end;

theorem :: RLVECT_2:56

for V being RealLinearSpace
for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 - L2 is Linear_Combination of A

proof end;

theorem Th57: :: RLVECT_2:57

for V being RealLinearSpace
for L being Linear_Combination of V holds L - L = ZeroLC V

proof end;

definition

let V be RealLinearSpace;

func LinComb V -> set means :Def14: :: RLVECT_2:def 14
for x being set holds
( x in it iff x is Linear_Combination of V );

proof end;

proof end;

end;

:: deftheorem Def14 defines LinComb RLVECT_2:def 14 :

registration

let V be RealLinearSpace;

cluster LinComb V -> non empty ;

proof end;

end;

definition

let V be RealLinearSpace;
let e be Element of LinComb V;

func @ e -> Linear_Combination of V equals :: RLVECT_2:def 15
e;

coherence by Def14;

end;

:: deftheorem defines @ RLVECT_2:def 15 :

definition

let V be RealLinearSpace;
let L be Linear_Combination of V;

func @ L -> Element of LinComb V equals :: RLVECT_2:def 16
L;

coherence by Def14;

end;

:: deftheorem defines @ RLVECT_2:def 16 :

definition

let V be RealLinearSpace;

func LCAdd V -> BinOp of (LinComb V) means :Def17: :: RLVECT_2:def 17
for e1, e2 being Element of LinComb V holds it . (e1,e2) = (@ e1) + (@ e2);

proof end;

proof end;

end;

:: deftheorem Def17 defines LCAdd RLVECT_2:def 17 :

definition

let V be RealLinearSpace;

func LCMult V -> Function of [:REAL,(LinComb V):],(LinComb V) means :Def18: :: RLVECT_2:def 18
for a being Real
for e being Element of LinComb V holds it . [a,e] = a * (@ e);

proof end;

proof end;

end;

:: deftheorem Def18 defines LCMult RLVECT_2:def 18 :

definition

let V be RealLinearSpace;

func LC_RLSpace V -> RLSStruct equals :: RLVECT_2:def 19
RLSStruct(# (LinComb V),(@ (ZeroLC V)),(LCAdd V),(LCMult V) #);

end;

:: deftheorem defines LC_RLSpace RLVECT_2:def 19 :

registration

let V be RealLinearSpace;

cluster LC_RLSpace V -> non empty strict ;

end;

registration

let V be RealLinearSpace;

cluster LC_RLSpace V -> right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ;

proof end;

end;

theorem :: RLVECT_2:58

for V being RealLinearSpace holds the carrier of (LC_RLSpace V) = LinComb V ;

theorem :: RLVECT_2:59

for V being RealLinearSpace holds 0. (LC_RLSpace V) = ZeroLC V ;

theorem :: RLVECT_2:60

for V being RealLinearSpace holds the addF of (LC_RLSpace V) = LCAdd V ;

theorem :: RLVECT_2:61

for V being RealLinearSpace holds the Mult of (LC_RLSpace V) = LCMult V ;

theorem Th62: :: RLVECT_2:62

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds (vector ((LC_RLSpace V),L1)) + (vector ((LC_RLSpace V),L2)) = L1 + L2

proof end;

theorem Th63: :: RLVECT_2:63

for V being RealLinearSpace
for a being Real
for L being Linear_Combination of V holds a * (vector ((LC_RLSpace V),L)) = a * L

proof end;

theorem Th64: :: RLVECT_2:64

for V being RealLinearSpace
for L being Linear_Combination of V holds - (vector ((LC_RLSpace V),L)) = - L

proof end;

theorem :: RLVECT_2:65

for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds (vector ((LC_RLSpace V),L1)) - (vector ((LC_RLSpace V),L2)) = L1 - L2

proof end;

definition

let V be RealLinearSpace;
let A be Subset of V;

func LC_RLSpace A -> strict Subspace of LC_RLSpace V means :: RLVECT_2:def 20
the carrier of it = { l where l is Linear_Combination of A : verum } ;

proof end;

uniqueness by RLSUB_1:30;

end;

:: deftheorem defines LC_RLSpace RLVECT_2:def 20 :

theorem Th66: :: RLVECT_2:66

for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)

proof end;

theorem :: RLVECT_2:67

for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)

proof end;

theorem :: RLVECT_2:68

for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for V being non empty right_complementable Abelian add-associative right_zeroed ModuleStr over R
for F, G, H being FinSequence of V st len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)

proof end;

theorem :: RLVECT_2:69

for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R holds a * (Sum (<*> the carrier of V)) = 0. V

proof end;

theorem :: RLVECT_2:70

for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for v, u being Element of V holds a * (Sum <*v,u*>) = (a * v) + (a * u)

proof end;

theorem :: RLVECT_2:71

for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)

proof end;