Codimension of intersection of subspaces
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Let $codim(E_i)=1$ for $i=1,...n$. Prove that $codim(cap E_i) le n$.
I was trying to prove it only with simple dimensions calculation - $codim(cap E_i) = dim(Esetminus cap E_i) = dim(Esetminus E_1 cap dots cap E_n) le dim(Esetminus E_1) + dots + dim(Esetminus E_i) = codim(E_1)+dots + codim(E_i) =n $
But are all my equations and inequalities correct?
If not, any other ideas?
linear-algebra hilbert-spaces dimension-theory
asked Nov 14 at 20:38
ChikChak
767418
add a comment |
up vote
0
down vote
favorite
Let $codim(E_i)=1$ for $i=1,...n$. Prove that $codim(cap E_i) le n$.
I was trying to prove it only with simple dimensions calculation - $codim(cap E_i) = dim(Esetminus cap E_i) = dim(Esetminus E_1 cap dots cap E_n) le dim(Esetminus E_1) + dots + dim(Esetminus E_i) = codim(E_1)+dots + codim(E_i) =n $
But are all my equations and inequalities correct?
If not, any other ideas?
linear-algebra hilbert-spaces dimension-theory
asked Nov 14 at 20:38
ChikChak
767418
1
Your second inequality is equality. The first one, I believe is true, can you tell why?
– Alonso Delfín
Nov 14 at 20:42
@Alonso Delfin Because if I'm not mistaken $dim(Ucap V)= dim(U)+dim(V) - dim(U+V) le dim(U) + dim(V)$?
– ChikChak
Nov 14 at 20:46
1
Yeah that works. There various ways to see it but essentially, your final goal is to verify that $dim(E/(Vcap U)) leq dim(E/V) + dim(E/U)$
– Alonso Delfín
Nov 14 at 21:27
@Alonso Delfin Could you share the other ways?
– ChikChak
Nov 14 at 21:28
1
From the top of my mind, I guess that I’d try to shot first that $E/(U cap V) subset E/U cap E/V$ and then the result will follow from the formula you gave.
– Alonso Delfín
Nov 14 at 21:34
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $codim(E_i)=1$ for $i=1,...n$. Prove that $codim(cap E_i) le n$.
I was trying to prove it only with simple dimensions calculation - $codim(cap E_i) = dim(Esetminus cap E_i) = dim(Esetminus E_1 cap dots cap E_n) le dim(Esetminus E_1) + dots + dim(Esetminus E_i) = codim(E_1)+dots + codim(E_i) =n $
But are all my equations and inequalities correct?
If not, any other ideas?
linear-algebra hilbert-spaces dimension-theory
asked Nov 14 at 20:38
ChikChak
767418
Let $codim(E_i)=1$ for $i=1,...n$. Prove that $codim(cap E_i) le n$.
I was trying to prove it only with simple dimensions calculation - $codim(cap E_i) = dim(Esetminus cap E_i) = dim(Esetminus E_1 cap dots cap E_n) le dim(Esetminus E_1) + dots + dim(Esetminus E_i) = codim(E_1)+dots + codim(E_i) =n $
But are all my equations and inequalities correct?
If not, any other ideas?
linear-algebra hilbert-spaces dimension-theory
linear-algebra hilbert-spaces dimension-theory
asked Nov 14 at 20:38
ChikChak
767418
asked Nov 14 at 20:38
ChikChak
767418
edited Nov 14 at 20:44
asked Nov 14 at 20:38
ChikChak
767418
asked Nov 14 at 20:38
ChikChak
767418
asked Nov 14 at 20:38
ChikChak
767418
767418
1
Your second inequality is equality. The first one, I believe is true, can you tell why?
– Alonso Delfín
Nov 14 at 20:42
@Alonso Delfin Because if I'm not mistaken $dim(Ucap V)= dim(U)+dim(V) - dim(U+V) le dim(U) + dim(V)$?
– ChikChak
Nov 14 at 20:46
1
Yeah that works. There various ways to see it but essentially, your final goal is to verify that $dim(E/(Vcap U)) leq dim(E/V) + dim(E/U)$
– Alonso Delfín
Nov 14 at 21:27
@Alonso Delfin Could you share the other ways?
– ChikChak
Nov 14 at 21:28
1
From the top of my mind, I guess that I’d try to shot first that $E/(U cap V) subset E/U cap E/V$ and then the result will follow from the formula you gave.
– Alonso Delfín
Nov 14 at 21:34
add a comment |
1
Your second inequality is equality. The first one, I believe is true, can you tell why?
– Alonso Delfín
Nov 14 at 20:42
@Alonso Delfin Because if I'm not mistaken $dim(Ucap V)= dim(U)+dim(V) - dim(U+V) le dim(U) + dim(V)$?
– ChikChak
Nov 14 at 20:46
1
Yeah that works. There various ways to see it but essentially, your final goal is to verify that $dim(E/(Vcap U)) leq dim(E/V) + dim(E/U)$
– Alonso Delfín
Nov 14 at 21:27
@Alonso Delfin Could you share the other ways?
– ChikChak
Nov 14 at 21:28
1
From the top of my mind, I guess that I’d try to shot first that $E/(U cap V) subset E/U cap E/V$ and then the result will follow from the formula you gave.
– Alonso Delfín
Nov 14 at 21:34
1
1
Your second inequality is equality. The first one, I believe is true, can you tell why?
– Alonso Delfín
Nov 14 at 20:42
Your second inequality is equality. The first one, I believe is true, can you tell why?
– Alonso Delfín
Nov 14 at 20:42
@Alonso Delfin Because if I'm not mistaken $dim(Ucap V)= dim(U)+dim(V) - dim(U+V) le dim(U) + dim(V)$?
– ChikChak
Nov 14 at 20:46
@Alonso Delfin Because if I'm not mistaken $dim(Ucap V)= dim(U)+dim(V) - dim(U+V) le dim(U) + dim(V)$?
– ChikChak
Nov 14 at 20:46
1
1
Yeah that works. There various ways to see it but essentially, your final goal is to verify that $dim(E/(Vcap U)) leq dim(E/V) + dim(E/U)$
– Alonso Delfín
Nov 14 at 21:27
Yeah that works. There various ways to see it but essentially, your final goal is to verify that $dim(E/(Vcap U)) leq dim(E/V) + dim(E/U)$
– Alonso Delfín
Nov 14 at 21:27
@Alonso Delfin Could you share the other ways?
– ChikChak
Nov 14 at 21:28
@Alonso Delfin Could you share the other ways?
– ChikChak
Nov 14 at 21:28
1
1
From the top of my mind, I guess that I’d try to shot first that $E/(U cap V) subset E/U cap E/V$ and then the result will follow from the formula you gave.
– Alonso Delfín
Nov 14 at 21:34
From the top of my mind, I guess that I’d try to shot first that $E/(U cap V) subset E/U cap E/V$ and then the result will follow from the formula you gave.
– Alonso Delfín
Nov 14 at 21:34
add a comment |
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1
Your second inequality is equality. The first one, I believe is true, can you tell why?
– Alonso Delfín
Nov 14 at 20:42
@Alonso Delfin Because if I'm not mistaken $dim(Ucap V)= dim(U)+dim(V) - dim(U+V) le dim(U) + dim(V)$?
– ChikChak
Nov 14 at 20:46
1
Yeah that works. There various ways to see it but essentially, your final goal is to verify that $dim(E/(Vcap U)) leq dim(E/V) + dim(E/U)$
– Alonso Delfín
Nov 14 at 21:27
@Alonso Delfin Could you share the other ways?
– ChikChak
Nov 14 at 21:28
1
From the top of my mind, I guess that I’d try to shot first that $E/(U cap V) subset E/U cap E/V$ and then the result will follow from the formula you gave.
– Alonso Delfín
Nov 14 at 21:34