Exercise 1.3.10

Question

Prove that $\mathrm{W}_{1}=\left\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathrm{F}^{n}: a_{1}+a_{2}+\cdots+a_{n}=0\right\}$ is a subspace of $\mathrm{F}^{n}$, but $\mathrm{W}_{2}=\left\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathrm{F}^{n}: a_{1}+a_{2}+\cdots+a_{n}=1\right\}$ is not.

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
We have $W_1$ is a subspace since it's a plane with normal vector $(1,1,\dots ,1)$. But this should be checked carefully. And since $0
otin W_2$, $W_2$ is not a subspace.
\end{proof}

Exercise 1.3.10

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Exercise 1.3.10

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