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Given that a basis is the smallest set of vectors that is linearly independent and spans a subspace, I am unsure if $u + v$ is in the set $\mathcal B$. If my understanding of span is not incorrect, span includes all linear combinations of its vectors within the span. However, $u + v$, if included in the basis, makes it linearly dependent which would make the statement false. Is my understanding of span incorrect?

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    $\begingroup$ Yes. What you said can be rephrased to "$u+v$ is in the span of B, but $u+v$ should not be in B$, which is correct. $\endgroup$ – lEm Mar 1 '19 at 7:27
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    $\begingroup$ No, you're correct. $\endgroup$ – Parcly Taxel Mar 1 '19 at 7:27
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Suppose $u+v $ is in $\mathcal B $. Then note that $1\cdot(u+v)-1\cdot u-1\cdot v+0\cdot (\text{other elements in $\mathcal B $})=0.$

But this contradicts the fact that the basis elements are linearly independent.

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$-1u-1v+1(u+v)=0$

But $-1,-1,1$ are not zero, so they aren’t linearly independent. Your idea is correct

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