Question on Subspaces

Let $V = R^n$ and define

U ={x∈V : x1 +x2 +···+xn =0}, W ={x∈V : x1 =x2 =···=x_nn}.

Show that the following statement holds:

(a)  For all $n≥2$, both U and W are subspaces of V and V = U⊕W.

(b) Find an example where this statement fails if we replace $V = R^n$ by $V = F^n$ for some other field $F$.

I'm trying to write it in a way that a clean, thought out proof should be written so I know how to write it for an upcoming test.


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  • Hi, sorry to be pedantic, but in your proof you state for n ≠ 0 whereas the question asks for n≥2. Would there be any way to include that? Or am I missing something? Thanks

  • To be precise, the hypothesis n≥2 is needed because otherwise U is 0. I'll make a note about that.

The answer is accepted.
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