Subspatii afine I

A = , −→
, Φ K
⊂
= ∅ −→
Φ | × ⊂ , , Φ| × K
−→ =
Φ | × ⊂
Φ A
×
, , Φ | ×
K
⊂
.

A = , −→
, Φ K
⊂
= ∅ −→
Φ | × ⊂ , , Φ| × K
−→ =
Φ | × ⊂
Φ A
×
, , Φ | ×
K
⊂
.

∅
= {} −→ = {}
(S, V, Φ) ⊂ S
π ⊂ S
V −→
−→
= | ⊂
−→ −→
π = | π ⊂ π
Φ : S × S → V π −→
−→ π
−→ = −→ π =
{ ∈ K | = } , ∈ M,(K), ∈ K
K
{ ∈ K | = }

K A = , −→
, Φ
K
A = , −→
, Φ
K A = , −→
, Φ
< <
K A = , −→
, Φ
−

K A =
, −→
, Φ ⊂
∈
−→
:= | ∈ −→
−→ =
∈ :=
−→
| ∈
−→
−→ =
−→
−→
⊂ ⇔ ∃ ∈ .. | ∈ ⊂ .
.
..

K A =
, −→
, Φ ⊂
∈
−→
:= | ∈ −→
−→ =
∈ :=
−→
| ∈
−→
−→ =
−→
−→
⊂ ⇔ ∃ ∈ .. | ∈ ⊂ .
.
..

K A =
, −→
, +
= ∅
∈ = + −→ + −→
:= + | ∈ −→
−→
⊂ = ∅ ∃ ∈ ∃ ⊂
..
= + ⊂
..
⊂
.. ∈ ¯ ∈ −→
= + [¯] [¯] −→
¯
π ⊂
.. ∈ π
¯, ¯ ∈ −→ π ,
π = + [¯, ¯] [¯, ¯] ¯
¯

K A =
, −→
, +
= ∅
∈ = + −→ + −→
:= + | ∈ −→
−→
⊂ = ∅ ∃ ∈ ∃ ⊂
..
= + ⊂
..
⊂
.. ∈ ¯ ∈ −→
= + [¯] [¯] −→
¯
π ⊂
.. ∈ π
¯, ¯ ∈ −→ π ,
π = + [¯, ¯] [¯, ¯] ¯
¯

K A =
, −→
, +
= ∅
∈ = + −→ + −→
:= + | ∈ −→
−→
⊂ = ∅ ∃ ∈ ∃ ⊂
..
= + ⊂
..
⊂
.. ∈ ¯ ∈ −→
= + [¯] [¯] −→
¯
π ⊂
.. ∈ π
¯, ¯ ∈ −→ π ,
π = + [¯, ¯] [¯, ¯] ¯
¯

K A =
, −→
, + ⊂ = ∅
K = ⊂
..
∀, ∈ ∀α ∈ K ⇒ α + ( − α) ∈
K = ⊂
..
∀, , ∈ ⇒ + + ∈
⊂ = ∅
⇔ =< >

K A =
, −→
, + ⊂ = ∅
K = ⊂
..
∀, ∈ ∀α ∈ K ⇒ α + ( − α) ∈
K = ⊂
..
∀, , ∈ ⇒ + + ∈
⊂ = ∅
⇔ =< >

⊂ = ∅ < >
, ∈
< , > = {α + ( − α) | α ∈ K}
< , > = + [ −→
] < >
< >
{, } < >
, , ∈
< , , > =
{α + β + ( − α − β) | α, β ∈ K} :=< >
< >= + [ −→
, −→
] {, , }
< >

⊂ = ∅ < >
, ∈
< , > = {α + ( − α) | α ∈ K}
< , > = + [ −→
] < >
< >
{, } < >
, , ∈
< , , > =
{α + β + ( − α − β) | α, β ∈ K} :=< >
< >= + [ −→
, −→
] {, , }
< >