V W K
V W
V W
dim V dim V.dim W W =
R
R
4
W
=<
α
1
= (1, ,1 0, 0);
α
2
= (0, ,1 1, 0);
α
3
= (0, ,0 1, 1) >
Z
=<
β
1
= (1, ,0 1, 0);
β
2
= (0, ,2 1, 1);
β >
3
= (1, 2, ,1 2)
W + Z W Z
R
4
/W
λx
1
+ x x x
2
+
3
+
4
= 1
x x x
1
+ λx
2
+
3
+
4
= 1
x
1
+ x x
2
+ λx
3
+
4
= 1
x
1
+ x
2
+ x
3
+ λx
4
= 1
x
1
, x , x , x
2 3 4
λ
V n End(V )
R
V
u, v End(V ) u v = 0
rank rank n(u) + (v)
u End(V )
v End(V ) u v = 0 rank rank n(u) + (v) =

Preview text:

V W K V ⊗ W V W dim V ⊗ W = dim V.dim W R R4 W =< − → α1 = (1, 1, 0, 0); − → α2 = (0, 1, 1, 0); − → α3 = (0, 0, 1, 1) > − → − → − →
Z =< β1 = (1, 0, 1, 0); β2 = (0, 2, 1, 1); β3 = (1, 2, 1, 2) > W + Z W ∩ Z R4/W  λx1 + x2 + x3 + x4 = 1    x1 + λx2 + x3 + x4 = 1 x1 + x2 + λx3 + x4 = 1    x1 + x2 + x3 + λx4 = 1 x1, x2, x3, x4 λ V n End(V ) R V u, v ∈ End(V ) u ◦ v = 0 rank(u) + rank(v) ≤ n u ∈ End(V ) v ∈ End(V ) u ◦ v = 0 rank(u) + rank(v) = n