Does a Subspace Have to Contain the Zero Vector?
The Formal Definition of a Subspace Is as Follows: It Must Contain the Zero-Vector. It Must Be Closed Under Addition: If V1∈S v 1 ∈ S and V2∈S v 2 ∈ S for Any...
The formal definition of a subspace is as follows: It must contain the zero-vector. It must be closed under addition: if v1∈S v 1 ∈ S and v2∈S v 2 ∈ S for any v1,v2 v 1 , v 2 , then it must be true that (v1+v2)∈S ( v 1 + v 2 ) ∈ S or else S is not a subspace.
Can a subspace not contain the zero vector?
If the set does not contain the zero vector, then it cannot be a subspace. For example, the set A in Example 1 above could not be a subspace of R 2 because it does not contain the vector 0 = (0, 0).
Why does a subspace need a zero vector?
It needs the zero vector because if there was no zero vector then it wouldn't be a vector space itself.
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