let u and v be two vector in a vector space V.show that  span (u,v)=span (u+2v,u-v)

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pramodpandey | College Teacher | (Level 3) Valedictorian

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Spans are sets, and as such, in order to prove they're equal, we show
`w inspan{u,v}<=>w in span{u+2v,u-v}`
``
Suppose `w in span{u,v}` . Then there exists real a and b such that
`w=au+bv` 
`w=((2a-b)/3)(u-v)+((a+b)/3)(u+2v)` 

( write au+bv=c(u-v)+d(u+2v) and solve equations by comparing coefficients for c and d . Expand, and you'll see that you get the original expression .)
 
Tthis proves that span{u,v} is a subset of span{u + 2v,u - v}.

Now, let's suppose we have a`w in span{u+2v,u-w}` .Then there exist a and b such that:
`w=a(u+2v)+b(u-v)`
`w=(a+b)u+(2a-b)v` 

Hence span{u + v, u - v} is a subset of span{u, v}. Since each is a subset of the other, they are equal.
 
span{u,v}=span{u+2v,u-v}
 
{`A sube B and B sube A => A=B `   }

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