Determine isotropic subspace

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Let $(V,beta)$ be a metric space with the symmetric bilinearform $beta$.



Let $V = F_7$



The corresponding structure matrix is
beginbmatrix0&0&5&0&0\0&0&3&1&0\5&3&2&1&3\0&1&1&0&0\0&0&3&0&0endbmatrix
First I'm asked to give a basis for $operatornameRad(V,beta)$ which is straightforward as such a Basis is given by $(5,0,0,0,1)$ for example.



Now comes the more complicated thing. The next step is to give a basis for $V_1$ where $(V_1,beta_1)$ is orthogonal to $operatornameRad(V,beta)$ and $beta_1$ is just the restriction of $beta$ to $V_1$ (as far as i understand the exercise).



Now for me the first attempt was to figure out that $V_1$ must have dimension 4. Then I could just pick 4 linear independent vectors which doesn't lie in $<(5,0,0,0,1)>$ for example $e_1,e_2,e_3,e_4$.



But with this attempt there has to be something wrong because if I want to determine the maximum isotropic subspace of $(V_1,beta_1)$ and use the basis of this subspace to form the hyberbolic basis of $V_1$ I get wrong results.



Is there anyone who could help me with this? Thank you!




linear-algebra




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    posted as math.stackexchange.com/questions/2857947/… then self deleted.
    – Will Jagy
    Jul 20 at 22:08














up vote
0
down vote

favorite
1












Let $(V,beta)$ be a metric space with the symmetric bilinearform $beta$.



Let $V = F_7$



The corresponding structure matrix is
beginbmatrix0&0&5&0&0\0&0&3&1&0\5&3&2&1&3\0&1&1&0&0\0&0&3&0&0endbmatrix
First I'm asked to give a basis for $operatornameRad(V,beta)$ which is straightforward as such a Basis is given by $(5,0,0,0,1)$ for example.



Now comes the more complicated thing. The next step is to give a basis for $V_1$ where $(V_1,beta_1)$ is orthogonal to $operatornameRad(V,beta)$ and $beta_1$ is just the restriction of $beta$ to $V_1$ (as far as i understand the exercise).



Now for me the first attempt was to figure out that $V_1$ must have dimension 4. Then I could just pick 4 linear independent vectors which doesn't lie in $<(5,0,0,0,1)>$ for example $e_1,e_2,e_3,e_4$.



But with this attempt there has to be something wrong because if I want to determine the maximum isotropic subspace of $(V_1,beta_1)$ and use the basis of this subspace to form the hyberbolic basis of $V_1$ I get wrong results.



Is there anyone who could help me with this? Thank you!




linear-algebra




share|cite|improve this question















  • 1




    posted as math.stackexchange.com/questions/2857947/… then self deleted.
    – Will Jagy
    Jul 20 at 22:08












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





Let $(V,beta)$ be a metric space with the symmetric bilinearform $beta$.



Let $V = F_7$



The corresponding structure matrix is
beginbmatrix0&0&5&0&0\0&0&3&1&0\5&3&2&1&3\0&1&1&0&0\0&0&3&0&0endbmatrix
First I'm asked to give a basis for $operatornameRad(V,beta)$ which is straightforward as such a Basis is given by $(5,0,0,0,1)$ for example.



Now comes the more complicated thing. The next step is to give a basis for $V_1$ where $(V_1,beta_1)$ is orthogonal to $operatornameRad(V,beta)$ and $beta_1$ is just the restriction of $beta$ to $V_1$ (as far as i understand the exercise).



Now for me the first attempt was to figure out that $V_1$ must have dimension 4. Then I could just pick 4 linear independent vectors which doesn't lie in $<(5,0,0,0,1)>$ for example $e_1,e_2,e_3,e_4$.



But with this attempt there has to be something wrong because if I want to determine the maximum isotropic subspace of $(V_1,beta_1)$ and use the basis of this subspace to form the hyberbolic basis of $V_1$ I get wrong results.



Is there anyone who could help me with this? Thank you!




linear-algebra




share|cite|improve this question











Let $(V,beta)$ be a metric space with the symmetric bilinearform $beta$.



Let $V = F_7$



The corresponding structure matrix is
beginbmatrix0&0&5&0&0\0&0&3&1&0\5&3&2&1&3\0&1&1&0&0\0&0&3&0&0endbmatrix
First I'm asked to give a basis for $operatornameRad(V,beta)$ which is straightforward as such a Basis is given by $(5,0,0,0,1)$ for example.



Now comes the more complicated thing. The next step is to give a basis for $V_1$ where $(V_1,beta_1)$ is orthogonal to $operatornameRad(V,beta)$ and $beta_1$ is just the restriction of $beta$ to $V_1$ (as far as i understand the exercise).



Now for me the first attempt was to figure out that $V_1$ must have dimension 4. Then I could just pick 4 linear independent vectors which doesn't lie in $<(5,0,0,0,1)>$ for example $e_1,e_2,e_3,e_4$.



But with this attempt there has to be something wrong because if I want to determine the maximum isotropic subspace of $(V_1,beta_1)$ and use the basis of this subspace to form the hyberbolic basis of $V_1$ I get wrong results.



Is there anyone who could help me with this? Thank you!





linear-algebra





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asked Jul 20 at 21:13









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  • 1




    posted as math.stackexchange.com/questions/2857947/… then self deleted.
    – Will Jagy
    Jul 20 at 22:08












  • 1




    posted as math.stackexchange.com/questions/2857947/… then self deleted.
    – Will Jagy
    Jul 20 at 22:08







1




1




posted as math.stackexchange.com/questions/2857947/… then self deleted.
– Will Jagy
Jul 20 at 22:08




posted as math.stackexchange.com/questions/2857947/… then self deleted.
– Will Jagy
Jul 20 at 22:08















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