Let `B = {v_1,...v_k}` be an independent set of vectors in a vector space `V` , and let `u in V` . Prove that if `u notin span(B)` then the set `{v_1,...,v_k,u}` is independent.

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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Suppose we have

`a_1v_1+a_2v_2+...+a_kv_k+a_{k+1}u=0.` We wish to show that all the coefficients `a_1,...,a_{k+1}` are zero. We begin by asking if `a_{k+1}` can be nonzero. Suppose it is. Then we can rearrange the equation to get

`a_{k+1}u=-a_1v_1-a_2v_2-...-a_nv_n,` and after dividing,

`u=-((a_1)/(a_{k+1}))v_1-...-((a_n)/(a_{k+1}))v_n,` which contradicts the fact that `u` is not in span(`B` ). Therefore we must have `a_{k+1}=0,` and our first equation reduces to

`a_1v_1+a_2v_2+...+a_kv_k=0.` But we know that the `v` 's are independent, and so `a_1,...,a_k` are all zero, along with `a_{k+1}.` This completes the proof.

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