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.
Web Ad1072
59
Answer
Answers can only be viewed under the following conditions:
 The questioner was satisfied with and accepted the answer, or
 The answer was evaluated as being 100% correct by the judge.
Alessandro Iraci
1.7K

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

Uh what do you mean? If n≥2 then n ≠ 0.


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.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
 answered
 395 views
 $60.00
Related Questions
 Linear algebra
 Find the general solution of the system of ODE $X'=\begin{bmatrix} 1 & 3 \\ 3 & 1 \end{bmatrix} X$
 Find where this discrete 3D spiral converges in explict terms
 Linear Algebra Help : Consider Two Planes, P1 and P2
 Find the values of a, for which the system is consistent. Give a geometric interpretation of the solution(s).
 Does $\sum_{n=2}^{\infty}\frac{\sin n}{n \ln n}$ converge or diverge?
 Allocation of Price and Volume changes to a change in Rate
 Consider the vector v = (3, 4, 5)^T, calculate the orthogonal projection