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1. ### Is this proof correct? If {u,v,w} is a basis for V then {u+v+w,v+w,w}is also a basis?

Thanks chiro, yes i know alternatives of proving the statement, i was just trying to give a proof with this alternative aproach. I know i can prove it by matrix properties, by linear independence of the vectors, and by proving <{ u , v , w }>=V<{ u+v+w , v+w , w }>. Thanks though.
2. ### Is this proof correct? If {u,v,w} is a basis for V then {u+v+w,v+w,w}is also a basis?

I thought it was implicit, that was my intention
3. ### Is this proof correct? If {u,v,w} is a basis for V then {u+v+w,v+w,w}is also a basis?

Thanks Hurkyl but isn't saying that for for any arbitrary x∈V=<{ u , v , w }> with d=a, d+e=b and d+e+f=c it implies the unique solution d=a, e=b-d=b-a , f=c-d-e=c-a-(b-a)=c-a-b+a=c-b ?
4. ### Is this proof correct? If {u,v,w} is a basis for V then {u+v+w,v+w,w}is also a basis?

Let u,v,w\in V a vector space over a field F such that u≠v≠w. If { u , v , w } is a basis for V. Prove that { u+v+w , v+w , w } is also a basis for V. Proof Let u,v,w\in V a vector space over a field F such that u≠v≠w. Let { u , v , w } be a basis for V. Because { u , v , w } its a basis...