Vector subspace definition

[diogomgf]

[MATH]F[/MATH] is a vector subspace of [MATH]E[/MATH] with [MATH]E[/MATH] being a vector space over [MATH]\mathbb{K}[/MATH]. If [MATH]u+v \in F[/MATH] and [MATH]u \in F[/MATH] why does [MATH]v \in F[/MATH] necessarily?

I know that [MATH]\forall u,v \in F; u+v \in F[/MATH].

 

[pka]

[MATH]F[/MATH] is a vector subspace of [MATH]E[/MATH] with [MATH]E[/MATH] being a vector space over [MATH]K[/MATH]. If [MATH]u+v \in F[/MATH] and [MATH]u \in F[/MATH] why does [MATH]v \in F[/MATH] necessarily?

Question: Does \(-u\in F~?\)

 

[diogomgf]

Question: Does \(-u\in F~?\)

Yes because [MATH]-1 \in \mathbb{K}; \space u \in F[/MATH].
By definition of subspace over [MATH]\mathbb{K}[/MATH] :[MATH] \space \space \forall a \in \mathbb{K}; \space u \in F; \space \space a\boxdot u \in F[/MATH]
Does that mean that [MATH]v \in F[/MATH] because [MATH]u+v[/MATH] respects the vector space adition and multiplication axioms and [MATH]v \in \mathbb{K}[/MATH]?

 

[pka]

So does that mean that \((u+v)+(-u)\in F~?\)

 

[diogomgf]

So does that mean that \((u+v)+(-u)\in F~?\)

I see... thank you.