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Can any set of vector containing n vector in a n dimensional vector space be used to generate all other possible vectors in that vector space?

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The answer is no. Consider the set $$\{(1,0),(2,0)\}$$ which is two distinct vectors in $\mathbb{R}^2$, and yet their span is not all of $\mathbb{R}^2$.

If you insist that the set of vectors be linearly independent, then the answer is yes, by the dimension theorem.

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