Exercise 1.3.29

Question

Let $F$ be a field that is not of characteristic two. Define
$$
\mathrm{W}_{1}=\left\{A \in \mathrm{M}_{n 	imes n}(F): A_{i j}=0 	ext { whenever } i \leq jight\}
$$
and $\mathrm{W}_{2}$ to be the set of all symmetric $n 	imes n$ matrices with entries from $F$. Both $\mathrm{W}_{1}$ and $\mathrm{W}_{2}$ are subspaces of $\mathrm{M}_{n 	imes n}(F)$. Prove that $\mathrm{M}_{n 	imes n}(F)=\mathrm{W}_{1} \oplus \mathrm{W}_{2}$. Compare this exercise with \href{./exercise_1-3-28}{Exercise 28}.

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
It's easy that $W_1\cap W_2=\{0\}$. And we have $$M_{n	imes n}(\mathbb{F})=\{A:A \in M_{n	imes n}(\mathbb{F})\}=\{(A-B(A))+B(A):A \in M_{n	imes n}(\mathbb{F})\}=W_1+ W_2,$$
where $B(A)$ is the matrix with $B_{ij}=B_{ji}=A_{ij}$ if $i\leq j$.
\end{proof}

Exercise 1.3.29

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Exercise 1.3.29

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