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Linear subspace/Intersection/Dimension estimate/Fact/Proof
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Linear subspace/Intersection/Dimension estimate/Fact
Proof
Due to
fact
, we have
dim
K
(
U
1
∩
U
2
)
=
dim
K
(
U
1
)
+
dim
K
(
U
2
)
−
dim
K
(
U
1
+
U
2
)
=
n
−
k
1
+
n
−
k
2
−
dim
K
(
U
1
+
U
2
)
≥
n
−
k
1
+
n
−
k
2
−
n
=
n
−
k
1
−
k
2
.
{\displaystyle {}{\begin{aligned}\dim _{K}{\left(U_{1}\cap U_{2}\right)}&=\dim _{K}{\left(U_{1}\right)}+\dim _{K}{\left(U_{2}\right)}-\dim _{K}{\left(U_{1}+U_{2}\right)}\\&=n-k_{1}+n-k_{2}-\dim _{K}{\left(U_{1}+U_{2}\right)}\\&\geq n-k_{1}+n-k_{2}-n\\&=n-k_{1}-k_{2}.\end{aligned}}}
To fact
