If you have a base of some subspace can you fill to the base of other bigger subspace?

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I have two subspaces $S,V$ of $W$. If I have a base of $S$ $b_1,b_2,...,b_k$ and $dimS<dimV$ can I fill this base to the base of $V$, I know that I can extend this base to a base of $W$, but can I do that for a base of $V$? For example base of V is $b_1,b_2,....,b_k,b_k+1,..b_l$






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    If $S$ is also a subspace of $V$ then yes, otherwise you cannot do this in general.
    – Dave
    Sep 1 at 4:51










  • yes S is subspace of V
    – Marko Å korić
    Sep 1 at 4:57










  • Then yeah: you can always extend the basis of a subspace to a basis for the whole space; in this case we just regard $V$ as the ambient space that $S$ is a subspace inside of.
    – Dave
    Sep 1 at 4:59














up vote
0
down vote

favorite












I have two subspaces $S,V$ of $W$. If I have a base of $S$ $b_1,b_2,...,b_k$ and $dimS<dimV$ can I fill this base to the base of $V$, I know that I can extend this base to a base of $W$, but can I do that for a base of $V$? For example base of V is $b_1,b_2,....,b_k,b_k+1,..b_l$






linear-algebra







share|cite|improve this question



















  • 1




    If $S$ is also a subspace of $V$ then yes, otherwise you cannot do this in general.
    – Dave
    Sep 1 at 4:51










  • yes S is subspace of V
    – Marko Å korić
    Sep 1 at 4:57










  • Then yeah: you can always extend the basis of a subspace to a basis for the whole space; in this case we just regard $V$ as the ambient space that $S$ is a subspace inside of.
    – Dave
    Sep 1 at 4:59












up vote
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up vote
0
down vote

favorite











I have two subspaces $S,V$ of $W$. If I have a base of $S$ $b_1,b_2,...,b_k$ and $dimS<dimV$ can I fill this base to the base of $V$, I know that I can extend this base to a base of $W$, but can I do that for a base of $V$? For example base of V is $b_1,b_2,....,b_k,b_k+1,..b_l$






linear-algebra







share|cite|improve this question















I have two subspaces $S,V$ of $W$. If I have a base of $S$ $b_1,b_2,...,b_k$ and $dimS<dimV$ can I fill this base to the base of $V$, I know that I can extend this base to a base of $W$, but can I do that for a base of $V$? For example base of V is $b_1,b_2,....,b_k,b_k+1,..b_l$






linear-algebra




linear-algebra






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited Sep 1 at 5:31









dmtri

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854317










asked Sep 1 at 4:48









Marko Škorić

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  • 1




    If $S$ is also a subspace of $V$ then yes, otherwise you cannot do this in general.
    – Dave
    Sep 1 at 4:51










  • yes S is subspace of V
    – Marko Å korić
    Sep 1 at 4:57










  • Then yeah: you can always extend the basis of a subspace to a basis for the whole space; in this case we just regard $V$ as the ambient space that $S$ is a subspace inside of.
    – Dave
    Sep 1 at 4:59












  • 1




    If $S$ is also a subspace of $V$ then yes, otherwise you cannot do this in general.
    – Dave
    Sep 1 at 4:51










  • yes S is subspace of V
    – Marko Å korić
    Sep 1 at 4:57










  • Then yeah: you can always extend the basis of a subspace to a basis for the whole space; in this case we just regard $V$ as the ambient space that $S$ is a subspace inside of.
    – Dave
    Sep 1 at 4:59







1




1




If $S$ is also a subspace of $V$ then yes, otherwise you cannot do this in general.
– Dave
Sep 1 at 4:51




If $S$ is also a subspace of $V$ then yes, otherwise you cannot do this in general.
– Dave
Sep 1 at 4:51












yes S is subspace of V
– Marko Å korić
Sep 1 at 4:57




yes S is subspace of V
– Marko Å korić
Sep 1 at 4:57












Then yeah: you can always extend the basis of a subspace to a basis for the whole space; in this case we just regard $V$ as the ambient space that $S$ is a subspace inside of.
– Dave
Sep 1 at 4:59




Then yeah: you can always extend the basis of a subspace to a basis for the whole space; in this case we just regard $V$ as the ambient space that $S$ is a subspace inside of.
– Dave
Sep 1 at 4:59















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