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1 −1 0 2
−1 2 0 1
2 −3 0 1
1 −1 0 2
−1 2 0 1
2 −3 0 1
−−−−−→
ℓ3 − 2ℓ1
−−−−→
ℓ2 + ℓ3
1 −1 0 2
−1 2 0 1
0 −1 0 −3
1 −1 0 2
0 1 0 3
0 0 0 0
−−−−→
ℓ2 + ℓ1
−−−−→
ℓ1 + ℓ2
1 −1 0 2
0 1 0 3
0 −1 0 −3
1 0 0 5
0 1 0 3
0 0 0 0
G = 〈(1, −1, 0, 2), (−1, 2, 0, 1), (2, −3, 0, 1)〉
= 〈(1, −1, 0, 2), (−1, 2, 0, 1), (0, −1, 0, −3)〉
= 〈(1, −1, 0, 2), (0, 1, 0, 3), (0, −1, 0, −3)〉
= 〈(1, −1, 0, 2), (0, 1, 0, 3), (0, 0, 0, 0)〉
= 〈(1, −1, 0, 2), (0, 1, 0, 3)〉
= 〈(1, 0, 0, 5), (0, 1, 0, 3)〉
dim G = 2
F + G = 〈(1, 1, 0, 0), (0, 0, 1, 0), (1, 0, 0, 5), (0, 1, 0, 3)〉
G
F +
1 0 0 5
0 1 0 3
0 1 2 0
0 0 0 1
−−−−→
ℓ2 − ℓ1
1 0 0 5
0 1 0 3
0 0 −2 0
0 0 0 1
dim (F + G) = dim F + dim G − dim (F ∩ G)
((1, 0, 0, 5), (0, 1, 0, 3), (0, 0, −2, 0), (0, 0, 0, 1))
¡ F + G
dim (F + G) = 4
dim (F ∩ G) = dim (F + G) − dim F − dim G = 0
F ∩ G = {0} ¨ ¢