# If $S$ is a subspace with basis $\mathcal B$, and $u$ and $v$ are in the set of $\mathcal B$, then $u + v$ is the set of $\mathcal B$.

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?

• 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. – lEm Mar 1 '19 at 7:27
• No, you're correct. – Parcly Taxel Mar 1 '19 at 7:27

Suppose $$u+v$$ is in $$\mathcal B$$. Then note that $$1\cdot(u+v)-1\cdot u-1\cdot v+0\cdot (\text{other elements in \mathcal B })=0.$$
$$-1u-1v+1(u+v)=0$$
But $$-1,-1,1$$ are not zero, so they aren’t linearly independent. Your idea is correct