1. ## Property of subspace

Hey,

Consider a subset $W$ of a vectorspace $V$. If I can show that $W$ is a subspace of $V$ would that imply that $W$ itself is a vectorspace?

Or do I have to go the other way and first show that the subset $W$ fullfills the conditions of a vectorspace and therefore is a subspace.

Cole someone kindly clarify.

Thanks.

2. Originally Posted by surjective
Hey,

Consider a subset $W$ of a vectorspace $V$. If I can show that $W$ is a subspace of $V$ would that imply that $W$ itself is a vectorspace?

Or do I have to go the other way and first show that the subset $W$ fullfills the conditions of a vectorspace and therefore is a subspace.

Cole someone kindly clarify.

Thanks.
By definition a subspace of a vector space is a subset of a vector space which is a vector space with the same operations as the ambient space. That said, since most of the axioms of a vector space are 'inherited' from the ambient space it suffices to prove closure under linear combinations to prove a subset of a vector space is a subspace.

3. ## properties of vectorspace

Hey,

Thanks. So if I can show that a non-empty subset is closed under addition and scalar multiplication then the subset would be a subspace and then also a vectorspace. right?

Thanks

4. Originally Posted by surjective
Hey,

Thanks. So if I can show that a non-empty subset is closed under addition and scalar multiplication then the subset would be a subspace and then also a vectorspace. right?

Thanks
Indeed.