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$q: mathbbR^n to mathbbR$ is a quadratic form. Find a subspace with maximum dimension that satisfies certain requirements.
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I see a lot of questions of the form
$q: mathbbR^n to mathbbR$ is a quadratic form. Find a subspace with maximum dimension that satisfies certain requirements.
where the requirements are e.g.
$$qleft(xright):?::0$$
where "?" is a placeholder for inequality or equality sign.
For example: find a maximum subspace of $mathbbR^n$ such that
$$qleft(xright):=::0$$
I always find the desired subspace with maximum dimension but I really want to know how to proof that this is indeed the subspace with maximum dimension.
Example:
$$qleft(x_1,x_2,x_3right)=beginpmatrixx_1&x_2&x_3endpmatrixbeginpmatrix1&2&-1\ 2&8&-4\ -1&-4&2endpmatrixbeginpmatrixx_1\ x_2\ x_3endpmatrix$$
I found that the desired subspace is $W=Spanleftbeginpmatrix0\ frac12\ 1endpmatrixright$
Of course that if there's another subspace $U$ with dimension 3 , then $U=mathbbR^3$ and this is easy to see a contradiction, but how to show that there's no subspace $U$ of $mathbbR^3$ with dimension 2 such that $qleft(xright)=0$ f.a. $xin U$?
linear-algebra
edited Aug 2 at 13:43
zzuussee
1,152419
asked Aug 2 at 12:58
idan di
33719
add a comment |Â
up vote
0
down vote
favorite
I see a lot of questions of the form
$q: mathbbR^n to mathbbR$ is a quadratic form. Find a subspace with maximum dimension that satisfies certain requirements.
where the requirements are e.g.
$$qleft(xright):?::0$$
where "?" is a placeholder for inequality or equality sign.
For example: find a maximum subspace of $mathbbR^n$ such that
$$qleft(xright):=::0$$
I always find the desired subspace with maximum dimension but I really want to know how to proof that this is indeed the subspace with maximum dimension.
Example:
$$qleft(x_1,x_2,x_3right)=beginpmatrixx_1&x_2&x_3endpmatrixbeginpmatrix1&2&-1\ 2&8&-4\ -1&-4&2endpmatrixbeginpmatrixx_1\ x_2\ x_3endpmatrix$$
I found that the desired subspace is $W=Spanleftbeginpmatrix0\ frac12\ 1endpmatrixright$
Of course that if there's another subspace $U$ with dimension 3 , then $U=mathbbR^3$ and this is easy to see a contradiction, but how to show that there's no subspace $U$ of $mathbbR^3$ with dimension 2 such that $qleft(xright)=0$ f.a. $xin U$?
linear-algebra
edited Aug 2 at 13:43
zzuussee
1,152419
asked Aug 2 at 12:58
idan di
33719
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I see a lot of questions of the form
$q: mathbbR^n to mathbbR$ is a quadratic form. Find a subspace with maximum dimension that satisfies certain requirements.
where the requirements are e.g.
$$qleft(xright):?::0$$
where "?" is a placeholder for inequality or equality sign.
For example: find a maximum subspace of $mathbbR^n$ such that
$$qleft(xright):=::0$$
I always find the desired subspace with maximum dimension but I really want to know how to proof that this is indeed the subspace with maximum dimension.
Example:
$$qleft(x_1,x_2,x_3right)=beginpmatrixx_1&x_2&x_3endpmatrixbeginpmatrix1&2&-1\ 2&8&-4\ -1&-4&2endpmatrixbeginpmatrixx_1\ x_2\ x_3endpmatrix$$
I found that the desired subspace is $W=Spanleftbeginpmatrix0\ frac12\ 1endpmatrixright$
Of course that if there's another subspace $U$ with dimension 3 , then $U=mathbbR^3$ and this is easy to see a contradiction, but how to show that there's no subspace $U$ of $mathbbR^3$ with dimension 2 such that $qleft(xright)=0$ f.a. $xin U$?
linear-algebra
edited Aug 2 at 13:43
zzuussee
1,152419
asked Aug 2 at 12:58
idan di
33719
I see a lot of questions of the form
$q: mathbbR^n to mathbbR$ is a quadratic form. Find a subspace with maximum dimension that satisfies certain requirements.
where the requirements are e.g.
$$qleft(xright):?::0$$
where "?" is a placeholder for inequality or equality sign.
For example: find a maximum subspace of $mathbbR^n$ such that
$$qleft(xright):=::0$$
I always find the desired subspace with maximum dimension but I really want to know how to proof that this is indeed the subspace with maximum dimension.
Example:
$$qleft(x_1,x_2,x_3right)=beginpmatrixx_1&x_2&x_3endpmatrixbeginpmatrix1&2&-1\ 2&8&-4\ -1&-4&2endpmatrixbeginpmatrixx_1\ x_2\ x_3endpmatrix$$
I found that the desired subspace is $W=Spanleftbeginpmatrix0\ frac12\ 1endpmatrixright$
Of course that if there's another subspace $U$ with dimension 3 , then $U=mathbbR^3$ and this is easy to see a contradiction, but how to show that there's no subspace $U$ of $mathbbR^3$ with dimension 2 such that $qleft(xright)=0$ f.a. $xin U$?
linear-algebra
edited Aug 2 at 13:43
zzuussee
1,152419
asked Aug 2 at 12:58
idan di
33719
edited Aug 2 at 13:43
zzuussee
1,152419
edited Aug 2 at 13:43
zzuussee
1,152419
edited Aug 2 at 13:43
zzuussee
1,152419
1,152419
asked Aug 2 at 12:58
idan di
33719
asked Aug 2 at 12:58
idan di
33719
asked Aug 2 at 12:58
idan di
33719
33719
add a comment |Â
add a comment |Â
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