$\dim (\ker f \cap \ker g) = n-2$

Question

Let $V$ a vector subspace of dimension $n$ on $\mathbb R$ and $f,g \in V^* \backslash \{0\}$ two linearly independent linear forms. I want to show that $\dim (\ker f \cap \ker g) = n-2$.

Since $f$ and $g$ are linear forms, I know that dim ker $f =n-1$ and dim ker $g =n-1$. I think I should use the fact that the two forms are linearly independent to $\dim (\ker f \cap \ker g) = n-2$ but I don't really see how...

I saw a proof with scalar product but I would like to see an alternative proof or maybe an explanation of the scalar product's proof.

Answer 1

If $f,g$ are linearly independent, $\ker f +\ker g=\mathbb{R}^n$.

$\dim(\ker f+\ker g) =\dim\ker f +\dim\ker g-\dim(\ker f\cap\ker g)$ implies that

$n = n-1+n-1-\dim(\ker f\cap\ker g)$, and $\dim(\ker f\cap\ker g)=n-2$.

Answer 2

Comments:

The following post contains the key part of the proof of your problem. Tsemo has mentioned it in his comment.

Proving that linear functionals are linearly independent if and only if their kernels are different

A bit more explanation/belaboring:

From this above post, you conclude that $\ker f \cap\ker g$ strictly contained in $\ker f$ and $\ker g$. Then we have the following: $$ \dim (\ker f + \ker g) = n-1 + (\dim\ker f -\dim (\ker f \cap\ker g).$$ Since $ (\ker f -\dim (\ker f \cap\ker g )>0$, we have $\dim(\ker f +\ker g)$ is strictly greater than $n-1$. This shows that $\ker f +\ker g = V.$ by the way, I assumed that $\dim (V) = n > 2$.

Answer 3

Another way:

Consider the linear maps $h : V \to \mathbb R^2$ and $\tilde h : \mathbb R^2 \to V^*$ given by $h(v) = (f(v),g(v))$ for all $v \in V$ and $\tilde h(a,b) = af+bg$ for all $(a,b) \in \mathbb R^2$. Prove that $h$ is surjective if $\tilde h$ is injective (hint: find an isomorphism $\psi : (\mathbb R^2)^* \to \mathbb R^2$ such that the tranpose of $h$ can be written as $\tilde h \circ \psi$). Thus, the linear independence of $f$ and $g$ shows that $h$ is surjective, and clearly $\ker h = \ker f \cap \ker g$. Finally, use the rank-nullity theorem on $h$.