Exercise 1.3.24

Question

Show that $\mathrm{F}^{n}$ is the direct sum of the subspaces
$$
\mathrm{W}_{1}=\left\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathrm{F}^{n}: a_{n}=0\right\}
$$
and
$$
\mathrm{W}_{2}=\left\{\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathrm{F}^{n}: a_{1}=a_{2}=\cdots=a_{n-1}=0\right\} .
$$

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
It's natural that $W_1\cap W_2=\{0\}$. And we have $\mathbb{F}^n=\{(a_1,a_2,\dots ,a_n):a_i\in \mathbb{F}\}=\{(a_1,a_2,\dots ,a_{n-1},0)+(0,0,\dots ,a_n):a_i\in \mathbb{F}\}=W_1\oplus W_2$.
\end{proof}

Exercise 1.3.24

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Exercise 1.3.24

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