Complement space of a finite dimensional space over a finite field
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Let $V$ be a finite dimensional space over the field $mathbb{F}_q$ of $q$ elements and let $Usubset V$ a subspace of $V$. How many subspaces $Wsubset V$ are there such that $Wcap U = 0 $ and $V=W+U$ ?
I've been trying to use Jordan canonical form but I think I'm missing something here, I just can't get it :(
linear-algebra finite-groups modules finite-fields jordan-normal-form
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
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add a comment |
$begingroup$
Let $V$ be a finite dimensional space over the field $mathbb{F}_q$ of $q$ elements and let $Usubset V$ a subspace of $V$. How many subspaces $Wsubset V$ are there such that $Wcap U = 0 $ and $V=W+U$ ?
I've been trying to use Jordan canonical form but I think I'm missing something here, I just can't get it :(
linear-algebra finite-groups modules finite-fields jordan-normal-form
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
$endgroup$
add a comment |
$begingroup$
Let $V$ be a finite dimensional space over the field $mathbb{F}_q$ of $q$ elements and let $Usubset V$ a subspace of $V$. How many subspaces $Wsubset V$ are there such that $Wcap U = 0 $ and $V=W+U$ ?
I've been trying to use Jordan canonical form but I think I'm missing something here, I just can't get it :(
linear-algebra finite-groups modules finite-fields jordan-normal-form
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
$endgroup$
Let $V$ be a finite dimensional space over the field $mathbb{F}_q$ of $q$ elements and let $Usubset V$ a subspace of $V$. How many subspaces $Wsubset V$ are there such that $Wcap U = 0 $ and $V=W+U$ ?
I've been trying to use Jordan canonical form but I think I'm missing something here, I just can't get it :(
linear-algebra finite-groups modules finite-fields jordan-normal-form
linear-algebra finite-groups modules finite-fields jordan-normal-form
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
asked Jan 6 at 10:15
Diego Andrés Jauré RomeroDiego Andrés Jauré Romero
205
205
add a comment |
add a comment |
1 Answer
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$begingroup$
Hint: count the number of ways that a given basis of $U$ can be completed into a basis of $V$, and divide by the number of bases of any fitting $W$.
answered Jan 6 at 10:30
MindlackMindlack
4,900211
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1 Answer
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$begingroup$
Hint: count the number of ways that a given basis of $U$ can be completed into a basis of $V$, and divide by the number of bases of any fitting $W$.
answered Jan 6 at 10:30
MindlackMindlack
4,900211
$endgroup$
add a comment |
$begingroup$
Hint: count the number of ways that a given basis of $U$ can be completed into a basis of $V$, and divide by the number of bases of any fitting $W$.
answered Jan 6 at 10:30
MindlackMindlack
4,900211
$endgroup$
add a comment |
$begingroup$
Hint: count the number of ways that a given basis of $U$ can be completed into a basis of $V$, and divide by the number of bases of any fitting $W$.
answered Jan 6 at 10:30
MindlackMindlack
4,900211
$endgroup$
Hint: count the number of ways that a given basis of $U$ can be completed into a basis of $V$, and divide by the number of bases of any fitting $W$.
answered Jan 6 at 10:30
MindlackMindlack
4,900211
answered Jan 6 at 10:30
MindlackMindlack
4,900211
answered Jan 6 at 10:30
MindlackMindlack
4,900211
answered Jan 6 at 10:30
MindlackMindlack
4,900211
4,900211
add a comment |
add a comment |
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