Article RLVECT_3, MML version 4.99.1005
:: RLVECT_3:th 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1 holds
Sum (b2 + b3) = (Sum b2) + Sum b3;
:: RLVECT_3:th 2
theorem
for b1 being Element of REAL
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Linear_Combination of b2 holds
Sum (b1 * b3) = b1 * Sum b3;
:: RLVECT_3:th 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Linear_Combination of b1 holds
Sum - b2 = - Sum b2;
:: RLVECT_3:th 4
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Linear_Combination of b1 holds
Sum (b2 - b3) = (Sum b2) - Sum b3;
:: RLVECT_3:attrnot 1 => RLVECT_3:attr 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
attr a2 is linearly-independent means
for b1 being Linear_Combination of a2
st Sum b1 = 0. a1
holds Carrier b1 = {};
end;
:: RLVECT_3:dfs 1
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is linearly-independent
it is sufficient to prove
thus for b1 being Linear_Combination of a2
st Sum b1 = 0. a1
holds Carrier b1 = {};
:: RLVECT_3:def 1
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
b2 is linearly-independent(b1)
iff
for b3 being Linear_Combination of b2
st Sum b3 = 0. b1
holds Carrier b3 = {};
:: RLVECT_3:attrnot 2 => RLVECT_3:attr 1
notation
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
antonym linearly-dependent for linearly-independent;
end;
:: RLVECT_3:th 6
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3 & b3 is linearly-independent(b1)
holds b2 is linearly-independent(b1);
:: RLVECT_3:th 7
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds not 0. b1 in b2;
:: RLVECT_3:th 8
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
{} the carrier of b1 is linearly-independent(b1);
:: RLVECT_3:exreg 1
registration
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
cluster linearly-independent Element of bool the carrier of a1;
end;
:: RLVECT_3:th 9
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
{b2} is linearly-independent(b1)
iff
b2 <> 0. b1;
:: RLVECT_3:th 10
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
{0. b1} is linearly-dependent(b1);
:: RLVECT_3:th 11
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1
st {b2,b3} is linearly-independent(b1)
holds b2 <> 0. b1 & b3 <> 0. b1;
:: RLVECT_3:th 12
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of the carrier of b1 holds
{b2,0. b1} is linearly-dependent(b1) & {0. b1,b2} is linearly-dependent(b1);
:: RLVECT_3:th 13
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
b2 <> b3 & {b2,b3} is linearly-independent(b1)
iff
b3 <> 0. b1 &
(for b4 being Element of REAL holds
b2 <> b4 * b3);
:: RLVECT_3:th 14
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of the carrier of b1 holds
b2 <> b3 & {b2,b3} is linearly-independent(b1)
iff
for b4, b5 being Element of REAL
st (b4 * b2) + (b5 * b3) = 0. b1
holds b4 = 0 & b5 = 0;
:: RLVECT_3:funcnot 1 => RLVECT_3:func 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
func Lin A2 -> strict Subspace of a1 means
the carrier of it = {Sum b1 where b1 is Linear_Combination of a2: TRUE};
end;
:: RLVECT_3:def 2
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being strict Subspace of b1 holds
b3 = Lin b2
iff
the carrier of b3 = {Sum b4 where b4 is Linear_Combination of b2: TRUE};
:: RLVECT_3:th 17
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of bool the carrier of b2 holds
b1 in Lin b3
iff
ex b4 being Linear_Combination of b3 st
b1 = Sum b4;
:: RLVECT_3:th 18
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b3 being Element of bool the carrier of b2
st b1 in b3
holds b1 in Lin b3;
:: RLVECT_3:th 19
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
Lin {} the carrier of b1 = (0). b1;
:: RLVECT_3:th 20
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st Lin b2 = (0). b1 & b2 <> {}
holds b2 = {0. b1};
:: RLVECT_3:th 21
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
for b3 being strict Subspace of b1
st b2 = the carrier of b3
holds Lin b2 = b3;
:: RLVECT_3:th 22
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 = the carrier of b1
holds Lin b2 = b1;
:: RLVECT_3:th 23
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
st b2 c= b3
holds Lin b2 is Subspace of Lin b3;
:: RLVECT_3:th 24
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1
st Lin b2 = b1 & b2 c= b3
holds Lin b3 = b1;
:: RLVECT_3:th 25
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1 holds
Lin (b2 \/ b3) = (Lin b2) + Lin b3;
:: RLVECT_3:th 26
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being Element of bool the carrier of b1 holds
Lin (b2 /\ b3) is Subspace of (Lin b2) /\ Lin b3;
:: RLVECT_3:th 27
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds ex b3 being Element of bool the carrier of b1 st
b2 c= b3 &
b3 is linearly-independent(b1) &
Lin b3 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);
:: RLVECT_3:th 28
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st Lin b2 = b1
holds ex b3 being Element of bool the carrier of b1 st
b3 c= b2 & b3 is linearly-independent(b1) & Lin b3 = b1;
:: RLVECT_3:modenot 1 => RLVECT_3:mode 1
definition
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
mode Basis of A1 -> Element of bool the carrier of a1 means
it is linearly-independent(a1) &
Lin it = RLSStruct(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1#);
end;
:: RLVECT_3:dfs 3
definiens
let a1 be non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct;
let a2 be Element of bool the carrier of a1;
To prove
a2 is Basis of a1
it is sufficient to prove
thus a2 is linearly-independent(a1) &
Lin a2 = RLSStruct(#the carrier of a1,the ZeroF of a1,the addF of a1,the Mult of a1#);
:: RLVECT_3:def 3
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1 holds
b2 is Basis of b1
iff
b2 is linearly-independent(b1) &
Lin b2 = RLSStruct(#the carrier of b1,the ZeroF of b1,the addF of b1,the Mult of b1#);
:: RLVECT_3:th 32
theorem
for b1 being non empty right_complementable strict Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st b2 is linearly-independent(b1)
holds ex b3 being Basis of b1 st
b2 c= b3;
:: RLVECT_3:th 33
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2 being Element of bool the carrier of b1
st Lin b2 = b1
holds ex b3 being Basis of b1 st
b3 c= b2;
:: RLVECT_3:th 35
theorem
for b1 being non empty set
for b2 being Choice_Function of b1
st not {} in b1
holds proj1 b2 = b1;
:: RLVECT_3:th 36
theorem
for b1 being set
for b2 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct holds
b1 in (0). b2
iff
b1 = 0. b2;
:: RLVECT_3:th 37
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b3
holds b2 /\ b4 is Subspace of b3;
:: RLVECT_3:th 38
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b3 & b2 is Subspace of b4
holds b2 is Subspace of b3 /\ b4;
:: RLVECT_3:th 39
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b3 & b4 is Subspace of b3
holds b2 + b4 is Subspace of b3;
:: RLVECT_3:th 40
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3, b4 being Subspace of b1
st b2 is Subspace of b3
holds b2 is Subspace of b3 + b4;
:: RLVECT_3:th 41
theorem
for b1 being non empty right_complementable Abelian add-associative right_zeroed RealLinearSpace-like RLSStruct
for b2, b3 being FinSequence of the carrier of b1
for b4 being Function-like quasi_total Relation of the carrier of b1,REAL holds
b4 (#) (b2 ^ b3) = (b4 (#) b2) ^ (b4 (#) b3);