# If u, v, w ∈ R n ,

If u, v, w ∈ R n , then span(u, v + w) = span(u + v, w)

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user_27qwe

Let u and v be vectors in ${\mathrm{ℝ}}^{n}$. span(u,v) is defined as the set of a linear combinations of vectors u and v

Given

$w\in span\left(u,v\right)$

This implies that the vector w can be written as a Linear Combination of vectors u and v

$w={c}_{1}u+{c}_{2}v$

Prove

$span\left(u,v\right)=span\left(u,v,w\right)$

Consider the RHS of the above equation

$span\left(u,v,w\right)={k}_{1}u+{k}_{2}v+{k}_{3}w$

Now, substitute w in terms of u and v implies

$span\left(u,v,w\right)={k}_{1}u+{k}_{2}v+{k}_{3}w\left[{c}_{1}u+{c}_{2}v\right]=\left({k}_{1}+{k}_{3}{c}_{1}\right)u+\left({k}_{2}+{k}_{3}{c}_{2}\right)v$

That is

$span\left(u,v,w\right)={K}_{1}u+{K}_{2}v=span\left(u,v\right)$