Linear independence of set in a vector space

Question

I have a vector space W, and a subspace $S \subseteq W$. If there is a set of linearly independent vectors such that $\{w1, ..., w_n\} \subset S$, is $\{w1, ..., w_n\}$ also linearly independent in V?

Answer 1

Yes, this is true, and follows immediately from the definition of linear (in)dependence. Your set of vectors do not care about any other vectors lying in a larger space. You've probably proven that e.g. $A =\{(1,0,0), (0,1,0), (0,0,1)\}$ is a set of linearly independent vectors in $\mathbb{R}^3$. Of course, $\mathbb{R}^3$ is a subspace of $\mathbb{R}^4$ and $\mathbb{R}^5$, and so on. It would be strange if for some of these superspaces $A$ would be dependent and in other superspaces independent.

Answer 2

Yes.

Reason: Recall the condition for linear independence is $$ c_1v_1 + \ldots + c_nv_n = \bf{0} \iff c_1 = \ldots = c_n= 0. $$

Since the scalar field of $V$ and $W$ are the same, picking any coefficients $c_1, \ldots, c_n$ other than $0$ will sum to a nonzero vector, regardless of whether we consider the vectors, $v_i$, as elements of $W$ or $V$.