撞，撞，撞！

小喇叭 MATH306 撞哂 PHYS121, 111，無奈之下只好揀 MATH321，將會有兩日對住同一個 professor 連續 3 個鐘- -。

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我 d linear algebra 真係好唔掂 (難怪我 B- = =)，證下面條 inequality 證左好耐 (仲要搵人幫拖)。($\mathcal L(X,Y)$ denotes a collection of linear maps from $X$ to $Y$.)

Problem 1. Let $U$ and $V$ be finite dimensional vector spaces, $S\in \mathcal L(V,W)$, $T\in \mathcal L(U,V)$, prove that $$\dim \text{Nul}(ST)\leq \dim \text{Nul}(S)+\dim \text{Nul}(T).$$
遲啲自己啲 fact prove 得夠多嘅話試下順便整埋 collection，而家煩惱在 d 書冇答案。

一條 analysis problem，可視為 202 練習：

Problem 2. Let $f$ be continuous on $[0,\pi]$, $n\in \mathbb{N}$. Prove that $$\lim_{n\to\infty}\int_{0}^{\pi}f(x)|\sin{nx}|\,dx=\frac{2}{\pi}\int_{0}^{\pi}f(x)\,dx.$$

冇記錯其實有 ${\displaystyle \lim_{n\to\infty}} \int_0^\pi f(x)g(nx)\,dx$ 之類嘅一般結果，但詳細唔記得左。