Yes/No : Is the quotient $\ker(g) / \ker(f)$ is zero-dimensional? > let $V$ be a real finite dimensional vector space and $f,g$ are nonzero linear functional on $V$ real vector space,Assume that $Ker(f)\subset Ker(g)$ Is statements is true/false ? > > Is $\frac{ker(g)}{ker(f)}\cong\mathbb{R}$? # My attempt : I think yes by rank Nullity theorem it is correct because $\dim kerf = \dim ker g$ so $\dim(\frac{ker(g)}{ker(f)})\cong dim(\mathbb{R})=1$ Is its true ?

The kernel of a non-zero linear form on an $n$-dimensional vector space is of dimension $n-1$ by the rank theorem. Hence both your kernels have the same dimension. As one kernel is included in the other they are equal. So the quotient zero.