begin
Lm1:
for n, m being Nat
for K being Field
for A being Matrix of n,m,K st ( n = 0 implies m = 0 ) & the_rank_of A = n holds
ex B being Matrix of m -' n,m,K st the_rank_of (A ^ B) = m
Lm2:
for n being Nat
for f being b1 -element real-valued FinSequence holds f is Point of (TOP-REAL n)
begin
definition
let n,
m be
Nat;
let M be
Matrix of
n,
m,
F_Real;
existence
( ( n <> 0 implies ex b1 being Function of (TOP-REAL n),(TOP-REAL m) st
for f being n -element real-valued FinSequence holds b1 . f = Line (((LineVec2Mx (@ f)) * M),1) ) & ( not n <> 0 implies ex b1 being Function of (TOP-REAL n),(TOP-REAL m) st
for f being n -element real-valued FinSequence holds b1 . f = 0. (TOP-REAL m) ) )
uniqueness
for b1, b2 being Function of (TOP-REAL n),(TOP-REAL m) holds
( ( n <> 0 & ( for f being n -element real-valued FinSequence holds b1 . f = Line (((LineVec2Mx (@ f)) * M),1) ) & ( for f being n -element real-valued FinSequence holds b2 . f = Line (((LineVec2Mx (@ f)) * M),1) ) implies b1 = b2 ) & ( not n <> 0 & ( for f being n -element real-valued FinSequence holds b1 . f = 0. (TOP-REAL m) ) & ( for f being n -element real-valued FinSequence holds b2 . f = 0. (TOP-REAL m) ) implies b1 = b2 ) )
correctness
consistency
for b1 being Function of (TOP-REAL n),(TOP-REAL m) holds verum;
;
end;
begin
Lm4:
for n, m being Nat
for M being Matrix of n,m,F_Real
for f being b1 -element real-valued FinSequence ex L being b2 -element FinSequence of REAL st
( |.((Mx2Tran M) . f).| <= (Sum L) * |.f.| & ( for i being Nat st i in dom L holds
L . i = |.(@ (Col (M,i))).| ) )