(0 )2 1436587@9BAC9EDGF4HI 7P5QHRFTSVU0DXW6F4H

¢ ¢
dim F = 3 1
1
1
0
0
1
0
1
1 2 −1 1
1 1 0 0
1 0 1 1
1 2 −1 1
−−−−→
ℓ2 − ℓ1
−−−−→
ℓ3 + ℓ2
1 1 0 0
0 −1 1 1
1 2 −1 1
1 1 0 0
0 −1 1 1
0 0 0 2
−−−−→
ℓ3 − ℓ1
−−−−→
1
2 ℓ3
1 1 0 0
0 −1 1 1
0 1 −1 1
1 1 0 0
0 −1 1 1
0 0 0 1
G = 〈 (1, 1, 0, 0), (0, −1, 1, 1), (0, 0, 0, 1) 〉 dim G = 3
F +
¡
G
F + G = 〈(1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1), (1, 1, 0, 0), (0, −1, 1, 1), (0, 0, 0, 1)〉
¢
¢
⎡
⎢
⎣
1 0 0 1
0 1 0 1
0 0 1 1
1 1 0 0
0 −1 1 1
0 0 0 1
⎤
⎥
⎦
−−−−→
ℓ4 − ℓ1
−−−−→
ℓ4 − ℓ2
−−−−−→
− 1
2 ℓ4
−−−−→
ℓ5 − ℓ4
⎡
⎢
⎣
⎡
⎢
⎣
⎡
⎢
⎣
⎡
⎢
⎣
1 0 0 1
0 1 0 1
0 0 1 1
0 1 0 −1
0 −1 1 1
0 0 0 1
1 0 0 1
0 1 0 1
0 0 1 1
0 0 0 −2
0 0 1 2
0 0 0 1
1 0 0 1
0 1 0 1
0 0 1 1
0 0 0 1
0 0 0 1
0 0 0 1
1 0 0 1
0 1 0 1
0 0 1 1
0 0 0 1
0 0 0 0
0 0 0 0
⎤
⎤
⎤
⎥
⎦ −−−−→
ℓ5 + ℓ2
⎥
⎦ −−−−→
ℓ5 − ℓ3
⎡
⎥
⎦ −−−−→ ⎢
ℓ6 − ⎢ ℓ5 ⎣
⎡
⎢
⎣
⎡
⎢
⎣
1 0 0 1
0 1 0 1
0 0 1 1
0 1 0 −1
0 0 1 2
0 0 0 1
1 0 0 1
0 1 0 1
0 0 1 1
0 0 0 −2
0 0 0 1
0 0 0 1
1 0 0 1
0 1 0 1
0 0 1 1
0 0 0 1
0 0 0 1
0 0 0 0
((1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1), (0, 0, 0, 1)) ¡ F + G
dim(F + G) 4 = dim(F ∩ G) = dim F + dim G − dim(F + G) =
2
⎤
⎥
⎦
⎤
⎥
⎦
⎤
⎥
⎦
⎤
⎥
⎦
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