Why Does a Subset of a Vector Space Need the Zero Vector to Be a Subspace?

[torquerotates]

I am curious as to why a subset of a vector space V must have the vector space V's zero vector be the subsets' zero vector in order to be a subspace. Its just not intuitive.

 

[HallsofIvy]

Why would it not be intuitive? A subspace, U, of vector space V must be closed under addition and scalar multiplication. If v is in subspace U, the (-1)v = -v is also. Then v+ (-v)= 0 is in U. That is, the zero vector of V, 0, is in U. But it is easy to show that the zero vector of a unique. Since 0 is in U and, of course, v+ 0= v for all v in U, there cannot be another zero vector in U.

 

[trambolin]

If each subspace has its own zero vector, then combine these subspaces in order to get a bigger subspace or even the whole space. We will get bunch of different zeros and the whole space will very entertaining, suddenly disappearing elements and discontinuities...

Also note that these are the rules of the game that are required to have, rather than anticipating their existence based on intuition.

 

[ejungkurth]

I am curious as to why a subset of a vector space V must have the vector space V's zero vector be the subsets' zero vector in order to be a subspace. Its just not intuitive.

What would you suggest as an alternative?

 

[John Creighto]

What would you suggest as an alternative?

A subspace which isn't a vector space?

 

[mrandersdk]

by definition a subspace have to be a vector space, and then all the other peoples arguments holds. What you are sugesting is just a simple subset, but that is not so interresting i linear algebra because it don't have the vector space properties.

by the way: The space in subspace means vectorspace, so it should really say subvectorspace. But a subset isn't a space so that's why there is no ambiguity.