I am trying to do the following differential topology exercise, I think I have come up with a solution but wanted to make sure it is correct
Let $A$ be a smooth submanifold of $\mathbb{R}^n$. Then for all $k<n$, almot all affine $k-$planes in $\mathbb{R}^n$ are transverse to $M$.
So the idea for this is to use the parametric transversality theorem. Let's focus on $k$-planes for a fixed $k$, we will have this will be generated by $v_1,...,v_k$ vectors. So let's consider the map $F: (\mathbb{R^n})^k\rightarrow C^{\infty}(\mathbb{R}^k,\mathbb{R}^n)$ that sends $(v_1,...,v_k)$ to $\sum_{i=1}^k\lambda_i v_i$. We will have that $F^{ev}:(\mathbb{R}^n)^k\times \mathbb{R}^k\rightarrow \mathbb{R}^n$ will be a submersion so that $F^{ev}\pitchfork M$, and it's a smooth map. So using the parametric transversality theorem we conclude that $\pitchfork (F,M)$ has full measure. So that almost all afine $k$-planes are transversal to $M$.
What do you guys think ? Thanks in advance.