Systèmes linéaires et matrices

k
ikRk
i1+i5=i2, i2=i3+i6, i3=i1+i4, i4=i5+i6.
R1i1−R5i5−R4i4=e, R5i5+R2i2+R6i6= 0,−R6i6+R3i3+R4i4= 0.
i1−i2+i5= 0
i2−i3−i6= 0
i1−i3+i4= 0
i4−i5−i6= 0
R1i1−R4i4−R5i5=e
R2i24 + R5i5+R6i6= 0
R3i3+R4i4+R6i6= 0
K4F eC6N6+H2SO4+H2O−→ K2SO4+F eSO4+N2H8SO4 + CO.
aK4F eC6N6+bH2SO4+cH2O−→ dK2SO4+eF eSO4+fN2H8SO4 + gCO.
K: 4a= 2d, F e :a=e, C : 6a=g,
N: 6a= 2f, H : 2b+ 2c= 8f, S :b=d+e+f,
O: 4b+c= 4d+ 4e+ 4f+g.

4a−2d= 0
a−e= 0
6a−g= 0
6a−2f= 0
2b+ 2c−8f= 0
b−d−e−f= 0
4b+c−4d−4e−4f−g= 0
(1,6,6,2,1,3,6)a
1K4F eC6N6+ 6H2SO4+ 6H2O−→ 2K2SO4+F eSO4+ 3N2H8SO4+6CO.
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KK
i ri1
r1=1
2r2+1 2r4
r2=1
3r1
r3=1
3r1+1 2r2+1 2r4
r4=1
3r1+r3

r1−1
2r2−1 2r4= 0
−1
3r1+r2= 0
−1
3r1−1 2r2+r3−1 2r4= 0
−1
3r1−r3+r4= 0
r1+r2+r3+r4= 1
3
13,1 13,4 13,5 13
Rnn−(x1, x2,··· , xn)
R2={(x, y)|x∈R, y ∈R}.
x1, . . . , xp
a1x1+. . . +apxp=b,
a1, . . . , apb
n p n
p p
(x1, x2, . . . , xp)∈Rpn
(x1, x2, . . . , xp)
3x1+ 2x2= 4
2x1+ 3x2= 1
(x1, x2) = (2,−1)
12x1+ 3x2= 4
24x1+ 6x2= 2
12x1+ 3x2= 4
12x1+ 3x2= 1 (L2←1
2L2)
4=1/2

3x1+ 2x2= 4
D
(x1,2−3
2x1)D
(0,2) + x11,−3
2,
D(0,2) (1,−3/2)
(S)n n
(S) : 
a11x1+a12x2+. . . +a1nxn=b1(L1)
a21x1+a22x2+. . . +a2nxn=b2(L2)
an1x1+an2x2+. . . +annxn=bn(Ln)
x1, x2, . . . , xn
(S)
(S) (S0)
LiLj(S)
(S) (S0)Li
λLiλ
(S) (S0)Li
Li+αLjj6=i α
Li
L1LiL1
Li
(S)⇐⇒ 
a11x1+a12x2+. . . +a1nxn=b1
a0
22x2+. . . +a0 2nxn=b0 2
a0
n2x2+. . . +a0
nnxn=b0 n
L1
(S)⇐⇒ 
a11x1+a12x2+a13x3+. . . +a1nxn=b1
a0
22x2+a0 23x3+. . . +a0 2nxn=b0 2
a00
33x3+. . . +a00 3nxn=b00 3
a(n−1)
nn xn=b(n−1) n

xn
xn−1x1
(S) : 
2x1+ 3x2+ 2x3= 1
x1+x2+ 2x3= 2
x1+ 2x2+x3= 4
(S)⇐⇒ 
2x1+ 3x2+ 2x3=1(L1←L2)
x1+x2+ 2x3=2(L2←L1)
x1+ 2x2+x3= 4
⇐⇒ 
x1+x2+ 2x3= 2
2x1+ 3x2+ 2x3=1(L2←L2−2L1)
x1+ 2x2+x3=4(L3←L3−L1)
⇐⇒ 
x1+x2+ 2x3= 2
x2−2x3=−3
x2−x3=2(L3←L3−L2)
⇐⇒ 
x1+x2+ 2x3= 2
x2−2x3=−3
x3= 5
⇐⇒ 
x1= 2 −7−2×5 = −15
x2=−3+2×5=7
x3= 5
S(−15,7,5)
(S) : 
x1+ 2x2+ 3x3= 1
4x1+ 5x2+ 6x3= 2
7x1+ 8x2+ 9x3= 4
(S)⇐⇒ 
x1+ 2x2+ 3x3= 1
4x1+ 5x2+ 6x3= 2 (L2←L2−4L1)
7x1+ 8x2+ 9x3=4(L3←L3−7L1)
⇐⇒ 
x1+ 2x2+ 3x3= 1
−3x2−6x3=−2
−6x2−12x3=−3 (L3←L3−2L2)
⇐⇒ 
x1+ 2x2+ 3x3= 1
−3x2−6x3=−2
0 = 1
0 = 1 (S)
(S) : 
x1+ 2x2+ 3x3= 1
4x1+ 5x2+ 6x3= 2
7x1+ 8x2+ 9x3= 3