Just needed help for this question.
If V is a Subspace. I know you can v1+v2∈V since V is a subspace but can you also say v1-v2∈V?
Thank you
Vector Subspace Elements
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vector-spaces
3 Answers
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$\mathbf{Hint}$: since $V $ is a subspace: if $v\in V $ then $cv\in V$ for any scalar $c $.
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Yes, because $V$ contains all linear combinations of its elements, so if $v_1,v_2 \in V$ then $v_1 - v_2 = v_1 + (-1)v_2 \in V$
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Yes. A subspace of a vector space contains ( by definition) all linear combinations of its elements. So, $$v_1 \in V \quad \mbox{and}\quad v_2 \in V \quad\Rightarrow \quad av_1+b v_2 \in V \quad \forall a,b \in \mathbb{K}$$
So also for $a=1$ and $b=-1$.