My question is same as the subject, well I was thinking of checking whether $\mathbb{R}^n_{+}=\{(x_1,x_2,\dots, x_n): x_i\ge 0\}$ is a vector space or not.
I think it is not a vector space due to the nonexistence of additive inverse of vectors, right? so I think there can not be any non-trivial vector subspace which will completely lie in $\mathbb{R}^n_{+}$
It is possible to construct a vector space structure on your set, but with the regular vector addition and scalar multiplication it is not a vector space for exactly the reason you mention.