let m be Nat; for K being Field
for W being strict Subspace of m -VectSp_over K st dim W < m holds
ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm A,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of A )
let K be Field; for W being strict Subspace of m -VectSp_over K st dim W < m holds
ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm A,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of A )
let W be strict Subspace of m -VectSp_over K; ( dim W < m implies ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm A,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of A ) )
assume A1:
dim W < m
; ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm A,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of A )
per cases
( dim W = 0 or dim W > 0 )
;
suppose A2:
dim W = 0
;
ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm A,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of A )then reconsider ONE =
1. K,
m as
Matrix of
m -' (dim W),
m,
K by NAT_D:40;
take
ONE
;
ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm ONE,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of ONE )take N =
Seg m;
( card N = m -' (dim W) & N c= Seg m & Segm ONE,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of ONE )A3:
len (1. K,m) = m
by MATRIX_1:25;
A4:
dim (Space_of_Solutions_of ONE) = 0
by Lm8;
A5:
m -' (dim W) = m
by A2, NAT_D:40;
A6:
width (1. K,m) = m
by MATRIX_1:25;
Space_of_Solutions_of ONE =
(Omega). (Space_of_Solutions_of ONE)
.=
(0). (Space_of_Solutions_of ONE)
by A4, VECTSP_9:33
.=
(0). W
by A6, VECTSP_4:48
.=
(Omega). W
by A2, VECTSP_9:33
.=
W
;
hence
(
card N = m -' (dim W) &
N c= Seg m &
Segm ONE,
(Seg (m -' (dim W))),
N = 1. K,
(m -' (dim W)) &
W = Space_of_Solutions_of ONE )
by A5, A3, A6, FINSEQ_1:78, MATRIX13:46;
verum end; suppose A7:
dim W > 0
;
ex A being Matrix of m -' (dim W),m,K ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm A,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of A )set ZERO =
0. K,
(m -' (dim W)),
m;
A8:
m - (dim W) > (dim W) - (dim W)
by A1, XREAL_1:11;
A9:
m -' (dim W) = m - (dim W)
by A1, XREAL_1:235;
then A10:
(
len (0. K,(m -' (dim W)),m) = m -' (dim W) &
width (0. K,(m -' (dim W)),m) = m )
by A8, MATRIX_1:24;
A11:
card (Seg m) = m
by FINSEQ_1:78;
consider A being
Matrix of
dim W,
m,
K,
N being
finite without_zero Subset of
NAT such that A12:
N c= Seg m
and A13:
dim W = card N
and A14:
Segm A,
(Seg (dim W)),
N = 1. K,
(dim W)
and
the_rank_of A = dim W
and A15:
lines A is
Basis of
W
by Th70;
set SA =
Segm A,
(Seg (dim W)),
((Seg m) \ N);
A16:
card ((Seg m) \ N) = (card (Seg m)) - (card N)
by A12, CARD_2:63;
then A17:
width (Segm A,(Seg (dim W)),((Seg m) \ N)) = m - (card N)
by A7, A11, MATRIX_1:24;
A18:
card (Seg (dim W)) = dim W
by FINSEQ_1:78;
then
len (Segm A,(Seg (dim W)),((Seg m) \ N)) = dim W
by A7, MATRIX_1:24;
then
width ((Segm A,(Seg (dim W)),((Seg m) \ N)) @ ) = dim W
by A13, A8, A17, MATRIX_2:12;
then A19:
width (- ((Segm A,(Seg (dim W)),((Seg m) \ N)) @ )) = dim W
by MATRIX_3:def 2;
A20:
card (Seg (m -' (dim W))) = m -' (dim W)
by FINSEQ_1:78;
then reconsider CC =
1. K,
(m -' (dim W)) as
Matrix of
card (Seg (m -' (dim W))),
card ((Seg m) \ N),
K by A1, A13, A16, A11, XREAL_1:235;
A21:
(
(Seg m) \ ((Seg m) \ N) = (Seg m) /\ N &
m -' (m -' (dim W)) = m - (m -' (dim W)) )
by NAT_D:35, XBOOLE_1:48, XREAL_1:235;
A22:
Indices (0. K,(m -' (dim W)),m) = [:(Seg (m -' (dim W))),(Seg m):]
by A9, A8, MATRIX_1:24;
then A23:
[:(Seg (m -' (dim W))),N:] c= Indices (0. K,(m -' (dim W)),m)
by A12, ZFMISC_1:118;
len ((Segm A,(Seg (dim W)),((Seg m) \ N)) @ ) = m - (dim W)
by A13, A8, A17, MATRIX_2:12;
then
len (- ((Segm A,(Seg (dim W)),((Seg m) \ N)) @ )) = m -' (dim W)
by A9, MATRIX_3:def 2;
then reconsider BB =
- ((Segm A,(Seg (dim W)),((Seg m) \ N)) @ ) as
Matrix of
card (Seg (m -' (dim W))),
card N,
K by A13, A20, A19, MATRIX_2:7;
A24:
N misses (Seg m) \ N
by XBOOLE_1:79;
A25:
(Seg m) \ N c= Seg m
by XBOOLE_1:36;
then A26:
[:(Seg (m -' (dim W))),((Seg m) \ N):] c= Indices (0. K,(m -' (dim W)),m)
by A22, ZFMISC_1:118;
[:(Seg (m -' (dim W))),N:] /\ [:(Seg (m -' (dim W))),((Seg m) \ N):] =
[:(Seg (m -' (dim W))),(N /\ ((Seg m) \ N)):]
by ZFMISC_1:122
.=
[:(Seg (m -' (dim W))),{} :]
by A24, XBOOLE_0:def 7
.=
{}
by ZFMISC_1:113
;
then
for
i,
j,
bi,
bj,
ci,
cj being
Nat st
[i,j] in [:(Seg (m -' (dim W))),N:] /\ [:(Seg (m -' (dim W))),((Seg m) \ N):] &
bi = ((Sgm (Seg (m -' (dim W)))) " ) . i &
bj = ((Sgm N) " ) . j &
ci = ((Sgm (Seg (m -' (dim W)))) " ) . i &
cj = ((Sgm ((Seg m) \ N)) " ) . j holds
BB * bi,
bj = CC * ci,
cj
;
then consider M being
Matrix of
m -' (dim W),
m,
K such that A27:
Segm M,
(Seg (m -' (dim W))),
N = BB
and A28:
Segm M,
(Seg (m -' (dim W))),
((Seg m) \ N) = CC
and
for
i,
j being
Nat st
[i,j] in (Indices M) \ ([:(Seg (m -' (dim W))),N:] \/ [:(Seg (m -' (dim W))),((Seg m) \ N):]) holds
M * i,
j = (0. K,(m -' (dim W)),m) * i,
j
by A10, A23, A26, Th9;
(Seg m) /\ N = N
by A12, XBOOLE_1:28;
then consider MV being
Matrix of
dim W,
m,
K such that A29:
Segm MV,
(Seg (dim W)),
N = 1. K,
(dim W)
and A30:
Segm MV,
(Seg (dim W)),
((Seg m) \ N) = - ((Segm M,(Seg (m -' (dim W))),N) @ )
and A31:
Lin (lines MV) = Space_of_Solutions_of M
by A7, A13, A9, A8, A16, A11, A28, A21, Th67, XBOOLE_1:36;
A32:
now A33:
Indices A = [:(Seg (dim W)),(Seg m):]
by A7, MATRIX_1:24;
let i,
j be
Nat;
( [i,j] in Indices A implies A * i,j = MV * i,j )assume A34:
[i,j] in Indices A
;
A * i,j = MV * i,jA35:
i in Seg (dim W)
by A34, A33, ZFMISC_1:106;
A36:
Indices A = Indices MV
by MATRIX_1:27;
A37:
rng (Sgm (Seg (dim W))) = Seg (dim W)
by FINSEQ_1:def 13;
dom (Sgm (Seg (dim W))) = Seg (dim W)
by A18, FINSEQ_3:45;
then consider x being
set such that A38:
x in Seg (dim W)
and A39:
(Sgm (Seg (dim W))) . x = i
by A35, A37, FUNCT_1:def 5;
reconsider x =
x as
Element of
NAT by A38;
A40:
j in Seg m
by A34, A33, ZFMISC_1:106;
now per cases
( j in N or not j in N )
;
suppose A41:
j in N
;
A * i,j = MV * i,jthen A42:
[i,j] in [:(Seg (dim W)),N:]
by A35, ZFMISC_1:106;
A43:
rng (Sgm N) = N
by A12, FINSEQ_1:def 13;
dom (Sgm N) = Seg (dim W)
by A12, A13, FINSEQ_3:45;
then consider y being
set such that A44:
y in Seg (dim W)
and A45:
(Sgm N) . y = j
by A41, A43, FUNCT_1:def 5;
reconsider y =
y as
Element of
NAT by A44;
A46:
[:(Seg (dim W)),N:] c= Indices A
by A12, A33, ZFMISC_1:118;
then A47:
[x,y] in Indices (Segm MV,(Seg (dim W)),N)
by A36, A37, A39, A43, A45, A42, MATRIX13:17;
[x,y] in Indices (Segm A,(Seg (dim W)),N)
by A37, A39, A43, A45, A46, A42, MATRIX13:17;
hence A * i,
j =
(Segm MV,(Seg (dim W)),N) * x,
y
by A14, A29, A39, A45, MATRIX13:def 1
.=
MV * i,
j
by A39, A45, A47, MATRIX13:def 1
;
verum end; suppose
not
j in N
;
MV * i,j = A * i,jthen A48:
j in (Seg m) \ N
by A40, XBOOLE_0:def 5;
then A49:
[i,j] in [:(Seg (dim W)),((Seg m) \ N):]
by A35, ZFMISC_1:106;
A50:
rng (Sgm ((Seg m) \ N)) = (Seg m) \ N
by A25, FINSEQ_1:def 13;
dom (Sgm ((Seg m) \ N)) = Seg (m -' (dim W))
by A13, A9, A16, A11, FINSEQ_3:45, XBOOLE_1:36;
then consider y being
set such that A51:
y in Seg (m -' (dim W))
and A52:
(Sgm ((Seg m) \ N)) . y = j
by A48, A50, FUNCT_1:def 5;
reconsider y =
y as
Element of
NAT by A51;
A53:
[:(Seg (dim W)),((Seg m) \ N):] c= Indices A
by A25, A33, ZFMISC_1:118;
then A54:
[x,y] in Indices (Segm A,(Seg (dim W)),((Seg m) \ N))
by A37, A39, A50, A52, A49, MATRIX13:17;
A55:
[x,y] in Indices (Segm MV,(Seg (dim W)),((Seg m) \ N))
by A36, A37, A39, A50, A52, A53, A49, MATRIX13:17;
then A56:
[x,y] in Indices ((Segm M,(Seg (m -' (dim W))),N) @ )
by A30, Lm1;
then A57:
[y,x] in Indices (Segm M,(Seg (m -' (dim W))),N)
by MATRIX_1:def 7;
then A58:
[y,x] in Indices ((Segm A,(Seg (dim W)),((Seg m) \ N)) @ )
by A27, Lm1;
thus MV * i,
j =
(- ((Segm M,(Seg (m -' (dim W))),N) @ )) * x,
y
by A30, A39, A52, A55, MATRIX13:def 1
.=
- (((Segm M,(Seg (m -' (dim W))),N) @ ) * x,y)
by A56, MATRIX_3:def 2
.=
- ((- ((Segm A,(Seg (dim W)),((Seg m) \ N)) @ )) * y,x)
by A27, A57, MATRIX_1:def 7
.=
- (- (((Segm A,(Seg (dim W)),((Seg m) \ N)) @ ) * y,x))
by A58, MATRIX_3:def 2
.=
((Segm A,(Seg (dim W)),((Seg m) \ N)) @ ) * y,
x
by RLVECT_1:30
.=
(Segm A,(Seg (dim W)),((Seg m) \ N)) * x,
y
by A54, MATRIX_1:def 7
.=
A * i,
j
by A39, A52, A54, MATRIX13:def 1
;
verum end; hence
A * i,
j = MV * i,
j
;
verum end; then reconsider lA =
lines MV as
Subset of
W by A15, MATRIX_1:28;
take
M
;
ex N being finite without_zero Subset of NAT st
( card N = m -' (dim W) & N c= Seg m & Segm M,(Seg (m -' (dim W))),N = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of M )take NN =
(Seg m) \ N;
( card NN = m -' (dim W) & NN c= Seg m & Segm M,(Seg (m -' (dim W))),NN = 1. K,(m -' (dim W)) & W = Space_of_Solutions_of M )
MV = A
by A32, MATRIX_1:28;
then
Lin lA = VectSpStr(# the
carrier of
W,the
addF of
W,the
ZeroF of
W,the
lmult of
W #)
by A15, VECTSP_7:def 3;
hence
(
card NN = m -' (dim W) &
NN c= Seg m &
Segm M,
(Seg (m -' (dim W))),
NN = 1. K,
(m -' (dim W)) &
W = Space_of_Solutions_of M )
by A1, A13, A16, A11, A28, A31, VECTSP_9:21, XBOOLE_1:36, XREAL_1:235;
verum end;