It is a problem from Guillemin and Pollack.
Suppose that $X$ is a submanifold of $\mathbb R^N$. Show that almost every vector space $V$ of any fixed dimension $l$ in $\mathbb R^N$ intersects $X$ transversally. [HINT: The set $S\subset (\mathbb R^N)^l$ consisting of all linearly independent $l$-tuples if vectors in $\mathbb R^N$ is open in $\mathbb R^{Nl}$, and the map $\mathbb R^l\times S\rightarrow \mathbb R^N$ defined by $[(t_1,\dots,t_l),v_1,\dots,v_l]\mapsto t_1v_1+\dots t_lv_l$ is a submersion.]
Suppose that $F:X\times S\to Y$ is a smooth map of manifolds and $Z$ is a submanifold of $Y$, all manifolds without boundary. If $F$ is transverse to $Z$ then for almost every $s\in S$ the map $f_s : x\mapsto F(x,s)$ is transverse to $Z$.
I have shown that the function in the hint is indeed a submersion and by theorem above,for every element the map is transversal and it is affine subspace of $\mathbb R^N$ Is it enough to prove it? I am not sure.