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Hi so this seems pretty intuitive but i dont see how i could formally prove it. Could i get some tips with this ?

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    Remember that linear independence of $v_1, \ldots, v_n$ means that the only scalars $a_1, \ldots, a_n$ that satisfy $a_1v_1 + \ldots + a_nv_n = 0$ is only when $a_1 = a_2 = \ldots = a_n = 0$. – 2017-01-12

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It's easier to prove the contrapositive: If the vectors of a set are linearly dependent, then when you add a vector, the new set will be linearly dependent.

Let's say $V$ is linearly dependent. If you can prove that $V+\{ \mathbf{v} \}$ is linearly dependent for any vector $\mathbf v$, you have proven the contrapositive of the original statement, which is logically equivalent to the original.

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Let $V$ be a set of linear independent vectors and $V_x$ be the same set without vector $x$. What can we say about $V$ if $V_x$ is linear dependent?

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    Oh so you mean i could say that if Vx was linear dependent then V would be linear dependent so its a contradiction ? – 2017-01-12
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    right! so $V_x$ must be linear independent. – 2017-01-12
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    thanks , how could i prove it by using the defintion that Mnifldz did ? – 2017-01-12