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Let vectors $v_1,v_2,v_3,v_4$ is base of space $V$, and if $W$ is subspace of $V$ such that $v_1,v_2\in W$ and $v_3,v_4\not \in W$ then $v_1,v_2$ is base of $W$?

My Professor said that you can not make a base of subspace from base of space, but you can make a base of space from base of subspace. But here I do not know how to prove, can you help me?

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No. Let $W$ be spanned by $v_1,v_2,v_3+v_4$. Can you verify teh following facts: $v_1,v_2 \in W, v_3 \notin W,v_4 \notin W$, and $v_1,v_2 ,v_3+v_4$ are independent?

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Since $v_1,v_2,v_3,v_4$ is base of space $V$, $W$ is subspace of $V$ and $v_1,v_2\in W$ we know that $\dim W\ge 2.$

But it is possible to have $\dim W=3.$ For example, assume $v_3+v_4\in W.$ That is, consider $W=span\{v_1,v_2,v_3+v_4\}$

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