I have started with differential topology and I try to solve exercises in the book Differential Topology by Victor Guillemin, Alan Pollack. There are 2 exercises in chapter Sard and Morse Theorem I do not know how to start:

  • Ex1. Let $X$ be a submanifold of $\mathbb{R}^n$. Prove that there exists a linear mapping $l: \mathbb{R}^n \to \mathbb{R}$ such that the restriction to $X$ is Morse function.

  • Ex2. Let $\phi: X \to \mathbb{R}^n$ be an immersion. Prove that for almost all $a_1,...,a_n$, $a_1 \phi_1 +...+a_n \phi_n$ is Morse function on $X$ where $\phi_1,...,\phi_n$ are the coordinate functions of $\phi$.

For Ex1, I know how to prove with a special case where $X= \mathbb{S}^{n-1}$ and explicit form of $l$ is the height function $(x_1,...x_n) \mapsto x_n$. But for the general case, I have no idea to start with. Any solution or hint is helpful.

Thank you in advance.


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