For $A\in M_n(F)$, prove that $(A^t)^t=A$.

Let $W\coloneqq\{(a_1,\ldots,a_n)\in F^n\::\:a_1+\cdots+a_n=1\}$ be given.

Is $W$ a subspace of $F^n$? Justify your answer.

Let $W_1$ be the subspace of all upper lower triangular $n\times n$ matrix over $F$.

1. Define a subset $W_2$ of $M_n(F)$ by

   $$ W_2\coloneqq \{A=(a_{ij})\in M_n(F)\::\:a_{ij}=0\text{ for all }i\ge j\} $$

   Show that $W_2\le M_n(F)$.

2. Prove that $M_n(F)=W_1\oplus W_2$.