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Let $w_1,\dots, w_l$ be vectors in a vector space $V$ and let $v_i ∈\text{span} \{w_1,\dots,w_l\}$ for all $i=1,2,\dots,k$. Find an example to show that it is not true that span$\{v_1,\dots,v_k\}$=span$\{w_1,\dots,w_l\}$. Under what conditions would span$\{v_1,\dots,v_k\}$=span$\{w_1,\dots,w_l\}$?

I'm stuck on how to get started with this. Any help is appreciated.

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    Well, for the counterexample, $v_i$ could be all the same vector (or all in the same one-dimensional subspace) and the $w_i$ could span more than two dimensions, but including the one-dimensional span of the $v$ – 2017-01-28

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Let $w_1 = (1,0,0)^T,\ w_2 = (0,1,0)^T,\ w_3 = (0,0,1)^T$. Then you can choose $v_1 = (1,0,0)^T,\ v_2 = (2,0,0)^T, v_3 =(3,0,0)^T$ and see that $span(v_1,v_2,v_3) \neq span(w_1, w_2, w_3)$. In fact $v_1, v_2, v_3$ form a line which pass through the orign and $w_1, w_2, w_3$ span $\mathbb{R}^3$.

Let $t$ be the number of linearly independent vectors contained in $W = \{w_1,\ldots w_k\}$. Then you have to choose $t$ linearly independent elements $v_1, \ldots v_t$ of $span(W)$ in order to have $span(\{v_1,\ldots,v_t\}) = span(W)$.