Vectors which combine or sum to a vector contained in a vector space are also in the vector space?

Question

From every definition of a vector space, V, it's clear that if $\mathbf v,\mathbf u \in V$ then $ \mathbf v+\mathbf u \in V$. But does the reverse implication also hold? i.e. is it the case that if $ \mathbf v+\mathbf u \in V$ then $\mathbf v, \mathbf u \in V$?

Similarly for any linear combination of vectors in V, is it the case that if $\alpha \mathbf v + \beta \mathbf u \in V$ then $\alpha \mathbf v, \beta \mathbf u \in V$ for any $\alpha, \beta \in \Bbb{R}$?

Answer 1

No, consider $E = \mathbb{R^2}$. Let $\vec{i} = (1,0)$ and denote by $V$ the vector space generated by $\vec{i}$. We have $V = \{(t,0), t \in \mathbb{R} \}$.

Now if $\vec{u} = (1,1)$ and $\vec{v} = (1,-1)$, then $\vec{u} + \vec{v} = (2,0)$, so that $\vec{u} + \vec{v} \in V$, yet $\vec{u} \notin V$ and $\vec{v} \notin V$.

Answer 2

The answer is no.

Consider the vector space of polynomials of degree less than or equal $3$.

$ x^4+x^2+1$ and $ -x^4+x^2+1$ are not in your vector space but if you add them the result $2x^2+2$ is in your vector space.