:: MATRIXR2 semantic presentation
AA48:
for F1, F2 being FinSequence of REAL
for j being Element of NAT st len F1 = len F2 holds
(F1 - F2) . j = (F1 . j) - (F2 . j)
theorem AF300: :: MATRIXR2:1
BB110:
for D being non empty set
for i being Element of NAT
for A being Matrix of D st 1 <= i & i <= len A holds
Line A,i = A . i
theorem BB112: :: MATRIXR2:2
theorem TTh4: :: MATRIXR2:3
theorem AA004: :: MATRIXR2:4
theorem TTh3: :: MATRIXR2:5
theorem AA160: :: MATRIXR2:6
theorem AA280: :: MATRIXR2:7
theorem :: MATRIXR2:8
TTh10:
for i, j being Element of NAT
for A being Matrix of REAL st len (- A) = len A & width (- A) = width A & [i,j] in Indices A holds
(- A) * i,j = - (A * i,j)
theorem Th11: :: MATRIXR2:9
theorem AA126b: :: MATRIXR2:10
theorem :: MATRIXR2:11
theorem Th13: :: MATRIXR2:12
theorem :: MATRIXR2:13
theorem AA222: :: MATRIXR2:14
theorem AA223: :: MATRIXR2:15
theorem AA224: :: MATRIXR2:16
theorem :: MATRIXR2:17
theorem AA222b: :: MATRIXR2:18
theorem AA223b: :: MATRIXR2:19
theorem AA224b: :: MATRIXR2:20
theorem :: MATRIXR2:21
theorem AA126: :: MATRIXR2:22
theorem AA225: :: MATRIXR2:23
theorem AA227: :: MATRIXR2:24
theorem AA230: :: MATRIXR2:25
theorem AA235: :: MATRIXR2:26
theorem :: MATRIXR2:27
theorem AA511b: :: MATRIXR2:28
theorem AA15: :: MATRIXR2:29
theorem AA002: :: MATRIXR2:30
theorem AF200: :: MATRIXR2:31
:: deftheorem defines Det MATRIXR2:def 1 :
theorem :: MATRIXR2:32
theorem Th4: :: MATRIXR2:33
theorem Th12: :: MATRIXR2:34
theorem AA234: :: MATRIXR2:35
for
D being non
empty set for
a1,
a2,
a3,
b1,
b2,
b3,
c1,
c2,
c3 being
Element of
D holds
<*<*a1,a2,a3*>,<*b1,b2,b3*>,<*c1,c2,c3*>*> is
Matrix of 3,
D
theorem AA125: :: MATRIXR2:36
theorem AA124: :: MATRIXR2:37
for
D being non
empty set for
A being
Matrix of 3,
D holds
A = <*<*(A * 1,1),(A * 1,2),(A * 1,3)*>,<*(A * 2,1),(A * 2,2),(A * 2,3)*>,<*(A * 3,1),(A * 3,2),(A * 3,3)*>*>
theorem :: MATRIXR2:38
for
A being
Matrix of 3,
REAL holds
Det A = (((((((A * 1,1) * (A * 2,2)) * (A * 3,3)) - (((A * 1,3) * (A * 2,2)) * (A * 3,1))) - (((A * 1,1) * (A * 2,3)) * (A * 3,2))) + (((A * 1,2) * (A * 2,3)) * (A * 3,1))) - (((A * 1,2) * (A * 2,1)) * (A * 3,3))) + (((A * 1,3) * (A * 2,1)) * (A * 3,2))
theorem Th1: :: MATRIXR2:39
A62:
idseq 0 is Permutation of Seg 0
by FINSEQ_2:65;
theorem Th3: :: MATRIXR2:40
theorem AA53: :: MATRIXR2:41
theorem :: MATRIXR2:42
theorem AA37: :: MATRIXR2:43
theorem :: MATRIXR2:44
theorem AA63: :: MATRIXR2:45
theorem AA219: :: MATRIXR2:46
theorem :: MATRIXR2:47
theorem :: MATRIXR2:48
theorem :: MATRIXR2:49
theorem :: MATRIXR2:50
theorem :: MATRIXR2:51
theorem :: MATRIXR2:52
theorem :: MATRIXR2:53
theorem :: MATRIXR2:54
theorem :: MATRIXR2:55
theorem AA514: :: MATRIXR2:56
theorem AA516: :: MATRIXR2:57
theorem AA524: :: MATRIXR2:58
theorem AA526: :: MATRIXR2:59
theorem :: MATRIXR2:60
for
n,
m,
k being
Element of
NAT for
B being
Matrix of
n,
m,
REAL for
A being
Matrix of
m,
k,
REAL st
n > 0 holds
for
i,
j being
Element of
NAT st
[i,j] in Indices (B * A) holds
(B * A) * i,
j = ((Line B,i) * A) . j
theorem AA527: :: MATRIXR2:61
theorem :: MATRIXR2:62
:: deftheorem defines 1_Rmatrix MATRIXR2:def 2 :
theorem AA202: :: MATRIXR2:63
theorem AA001: :: MATRIXR2:64
theorem AA290: :: MATRIXR2:65
theorem :: MATRIXR2:66
theorem AA3661b: :: MATRIXR2:67
theorem AA3662b: :: MATRIXR2:68
theorem :: MATRIXR2:69
theorem AA500b: :: MATRIXR2:70
theorem AA501: :: MATRIXR2:71
theorem AA220: :: MATRIXR2:72
:: deftheorem defines 0_Rmatrix MATRIXR2:def 3 :
theorem :: MATRIXR2:73
:: deftheorem defines Base_FinSeq MATRIXR2:def 4 :
theorem AA1100: :: MATRIXR2:74
theorem AA1110: :: MATRIXR2:75
theorem AA1120: :: MATRIXR2:76
theorem :: MATRIXR2:77
(
Base_FinSeq 1,1
= <*1*> &
Base_FinSeq 2,1
= <*1,0*> &
Base_FinSeq 2,2
= <*0,1*> &
Base_FinSeq 3,1
= <*1,0,0*> &
Base_FinSeq 3,2
= <*0,1,0*> &
Base_FinSeq 3,3
= <*0,0,1*> )
theorem AA1200: :: MATRIXR2:78
:: deftheorem AA640 defines invertible MATRIXR2:def 5 :
:: deftheorem AA444 defines Inv MATRIXR2:def 6 :
theorem :: MATRIXR2:79
theorem AA1320: :: MATRIXR2:80
theorem :: MATRIXR2:81
theorem :: MATRIXR2:82
theorem :: MATRIXR2:83
theorem :: MATRIXR2:84
theorem :: MATRIXR2:85
theorem AA3924: :: MATRIXR2:86
theorem AA990: :: MATRIXR2:87
theorem AA3924a: :: MATRIXR2:88
theorem AA1000: :: MATRIXR2:89
theorem :: MATRIXR2:90
theorem :: MATRIXR2:91
theorem :: MATRIXR2:92
theorem AA1300: :: MATRIXR2:93
theorem :: MATRIXR2:94
theorem AA53: :: MATRIXR2:95
theorem AA1310: :: MATRIXR2:96
theorem :: MATRIXR2:97