Intersection of subspaces
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I am working on an example of vector spaces. I have the following question:
Let ${V_1,V_2,ldots,V_t}$ be a family of $n$-dimensional subspaces and the dimension intersection of any $n$ subspace is at least one. Is it true that
$dim bigcap^{i=t}_{i=1}V_igeq1$?
I have calculated $dim bigcap^{i=t}_{i=1}V_i$ for several spaces and all of spaces satisfied in $dim bigcap^{i=t}_{i=1}V_igeq1$.
Can anybody take counterexample?
linear-algebra vector-spaces
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
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add a comment |
$begingroup$
I am working on an example of vector spaces. I have the following question:
Let ${V_1,V_2,ldots,V_t}$ be a family of $n$-dimensional subspaces and the dimension intersection of any $n$ subspace is at least one. Is it true that
$dim bigcap^{i=t}_{i=1}V_igeq1$?
I have calculated $dim bigcap^{i=t}_{i=1}V_i$ for several spaces and all of spaces satisfied in $dim bigcap^{i=t}_{i=1}V_igeq1$.
Can anybody take counterexample?
linear-algebra vector-spaces
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
$endgroup$
$begingroup$
Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections?
$endgroup$
– Sigur
Jun 30 '12 at 21:03
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@Sigur yes exactly
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:06
add a comment |
$begingroup$
I am working on an example of vector spaces. I have the following question:
Let ${V_1,V_2,ldots,V_t}$ be a family of $n$-dimensional subspaces and the dimension intersection of any $n$ subspace is at least one. Is it true that
$dim bigcap^{i=t}_{i=1}V_igeq1$?
I have calculated $dim bigcap^{i=t}_{i=1}V_i$ for several spaces and all of spaces satisfied in $dim bigcap^{i=t}_{i=1}V_igeq1$.
Can anybody take counterexample?
linear-algebra vector-spaces
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
$endgroup$
I am working on an example of vector spaces. I have the following question:
Let ${V_1,V_2,ldots,V_t}$ be a family of $n$-dimensional subspaces and the dimension intersection of any $n$ subspace is at least one. Is it true that
$dim bigcap^{i=t}_{i=1}V_igeq1$?
I have calculated $dim bigcap^{i=t}_{i=1}V_i$ for several spaces and all of spaces satisfied in $dim bigcap^{i=t}_{i=1}V_igeq1$.
Can anybody take counterexample?
linear-algebra vector-spaces
linear-algebra vector-spaces
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
edited Dec 26 '18 at 12:08
Babak Miraftab
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
asked Jun 30 '12 at 20:57
Babak MiraftabBabak Miraftab
5,38212148
5,38212148
$begingroup$
Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections?
$endgroup$
– Sigur
Jun 30 '12 at 21:03
$begingroup$
@Sigur yes exactly
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:06
add a comment |
$begingroup$
Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections?
$endgroup$
– Sigur
Jun 30 '12 at 21:03
$begingroup$
@Sigur yes exactly
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:06
$begingroup$
Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections?
$endgroup$
– Sigur
Jun 30 '12 at 21:03
$begingroup$
Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections?
$endgroup$
– Sigur
Jun 30 '12 at 21:03
$begingroup$
@Sigur yes exactly
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:06
$begingroup$
@Sigur yes exactly
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:06
add a comment |
1 Answer
1
active
oldest
votes
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Consider the vector space $mathbb{R}^3$. Let $t = 3$ and $n = 2$. Consider $V_1$ to be the $xy$ plane. $V_2$ to be the $xz$ plane. $V_3$ to be the $yz$ plane. All of these are $2$ dimensional subspaces. The intersection of any pair is one dimensional line. However the intersection of all three is just the origin which is not $1$-dimensional.
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
$endgroup$
$begingroup$
ok thanks,it is very helpful
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:07
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
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votes
$begingroup$
Consider the vector space $mathbb{R}^3$. Let $t = 3$ and $n = 2$. Consider $V_1$ to be the $xy$ plane. $V_2$ to be the $xz$ plane. $V_3$ to be the $yz$ plane. All of these are $2$ dimensional subspaces. The intersection of any pair is one dimensional line. However the intersection of all three is just the origin which is not $1$-dimensional.
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
$endgroup$
$begingroup$
ok thanks,it is very helpful
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:07
add a comment |
$begingroup$
Consider the vector space $mathbb{R}^3$. Let $t = 3$ and $n = 2$. Consider $V_1$ to be the $xy$ plane. $V_2$ to be the $xz$ plane. $V_3$ to be the $yz$ plane. All of these are $2$ dimensional subspaces. The intersection of any pair is one dimensional line. However the intersection of all three is just the origin which is not $1$-dimensional.
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
$endgroup$
$begingroup$
ok thanks,it is very helpful
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:07
add a comment |
$begingroup$
Consider the vector space $mathbb{R}^3$. Let $t = 3$ and $n = 2$. Consider $V_1$ to be the $xy$ plane. $V_2$ to be the $xz$ plane. $V_3$ to be the $yz$ plane. All of these are $2$ dimensional subspaces. The intersection of any pair is one dimensional line. However the intersection of all three is just the origin which is not $1$-dimensional.
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
$endgroup$
Consider the vector space $mathbb{R}^3$. Let $t = 3$ and $n = 2$. Consider $V_1$ to be the $xy$ plane. $V_2$ to be the $xz$ plane. $V_3$ to be the $yz$ plane. All of these are $2$ dimensional subspaces. The intersection of any pair is one dimensional line. However the intersection of all three is just the origin which is not $1$-dimensional.
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
answered Jun 30 '12 at 21:03
WilliamWilliam
17.3k22256
17.3k22256
$begingroup$
ok thanks,it is very helpful
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:07
add a comment |
$begingroup$
ok thanks,it is very helpful
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:07
$begingroup$
ok thanks,it is very helpful
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:07
$begingroup$
ok thanks,it is very helpful
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:07
add a comment |
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$begingroup$
Each $V_i$ is $n$-dimensional. But you are taking exactly $n$ to take intersections?
$endgroup$
– Sigur
Jun 30 '12 at 21:03
$begingroup$
@Sigur yes exactly
$endgroup$
– Babak Miraftab
Jun 30 '12 at 21:06