(0 )2 1436587@9BAC9EDGF4HI 7P5QHRFTSVU0DXW6F4H

α = 1 β = 1 ¡ ¢ 1
1 1 1
0 −1 0
0 0 0
x
y
z
=
1
−1
0
y = 1 x + z = 0
{(x, 1, −x) | x ∈ IR}
⇐⇒
x + y + z
−y
0
=
1
−1
0
161
¢¡
(a, b, c, d) ∈ F£¢ c = d = a + b F ¢
F = {(a, b, a + b, a + b) | a, b ∈ IR}
= {a(1, 0, 1, 1) + b(0, 1, 1, 1) | a, b ∈ IR}
= 〈 (1, 0, 1, 1), (0, 1, 1, 1) 〉.
¢
§ ¢ ¡
1 0 1 1
0 1 1 1
((1, 0, 1, 1), (0, 1, 1, 1)) F dim F = 2
1 0 1 1
2 0 2 4
0 0 0 1
¢ ¢
1 0 1 1
2 0 2 4
0 0 0 1
−−−−−→
ℓ2 − 2ℓ1
1 0 1 1
0 0 0 2
0 0 0 1
−−−−→
1
2 ℓ2
1 0 1 1
0 0 0 1
0 0 0 1
−−−−→
ℓ2 − ℓ3
G = 〈 (1, 0, 1, 1), (0, 0, 0, 1) 〉 dim G = 2
(a, b, c, d) ∈ F ∩ G£¢ (a, b, c, d) ∈ F (a, b, c, d) ∈ G
(a, b, c, d) = (a, b, a + b, a + b)
(a, b, c, d) = α(1, 0, 1, 1) + β(0, 0, 0, 1) = (α, 0, α, α + β) =⇒
1 0 1 1
0 0 0 1
0 0 0 0
¡
α = a
F ∩ G = {(a, 0, a, a) | a ∈ IR} = 〈(1, 0, 1, 1)〉 dim F ∩ G = 1
dim(F + G) = dim F + dim G − dim F ∩ G = 3
β = 0
b = 0