If u, v, w ∈ R n ,

2022-03-30 Answered

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

 

 

 

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Answers (1)

user_27qwe
Answered 2022-05-13 Author has 208 answers

Let u and v be vectors in n. span(u,v) is defined as the set of a linear combinations of vectors u and v

 

Given

wspan(u,v)

 

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

w=c1u+c2v

 

Prove

span(u,v)=span(u,v,w)

 

Consider the RHS of the above equation

span(u,v,w)=k1u+k2v+k3w

 

Now, substitute w in terms of u and v implies

span(u,v,w)=k1u+k2v+k3w[c1u+c2v]=(k1+k3c1)u+(k2+k3c2)v

 

That is

span(u,v,w)=K1u+K2v=span(u,v)

 

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