$\mathbf {The \ Problem \ is}:$ If $X$ is an embedded submanifold in $\mathbb R^N$ ,show there exists a linear map $L : \mathbb R^N \to \mathbb R$ such that $L|_X$ is a Morse map .
$\mathbf {My \ approach}:$ Exc $14$ is a special case with $X=S^{N-1}$ and $L=π_N$ , so I think for general case we can take $L=π_N$ $=> x\in X$ is critical point iff $T_xX= \mathbb R^{N-1}×\{0\}$ but after that I can't approach . A hint is appreciated, thanks in advance .
You can simply use the theorem from page 43 in the Guillemin-Pollack book, take the smooth function $f$ to be zero function, then the theorem tells us that for almost every $a \in \mathbb R^N$，the function $f = \sum_{i=1}^{N}a_{i}x_{i}$ is a Morse function on X, take the linear map $L = \sum_{i=1}^{N}a_{i}x_{i}$ to satisfy the condition.