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
- 355 views
- $60.00
Related Questions
- How to filter data with the appearance of a Sine wave to 'flattern' the peaks
- Conjugate / Transpose - Matrix
- Elementary row reduction for an $n\times n$ matrix
- Allocation of Price and Volume changes to a change in Rate
- [Linear Algebra] $T$-invariant subspace
- Find $x$ so that $\begin{pmatrix} 1 & 0 & c \\ 0 & a & -b \\ -\frac{1}{a} & x & x^2 \end{pmatrix}$ is invertible
- Prove that $V={(𝑥_1,𝑥_2,⋯,𝑥_n) \in ℝ^n ∣ 𝑥_1+𝑥_2+...+𝑥_{𝑛−1}−2𝑥_𝑛=0}\}$ is a subspace of $\R^n$.
- Sum of column spaces