Linear Algebra Done Right · Problem 3E.16

A linear functional is a map into the scalar field, so its image is either $\{0\}$ or all of $\mathbf{F}$—and since $\varphi \neq 0$, it must be the whole field. The rank-nullity theorem then tells you that the dimension of the quotient space $V/(\operatorname{null} \varphi)$ equals the dimension of the image, which is $\dim \mathbf{F} = 1$.

A linear functional is a map into the scalar field, so its image is either $\{0\}$ or all of $\mathbf{F}$—and since $\varphi \neq 0$, it must be the whole field. The rank-nullity theorem then tells you that the dimension of the quotient space $V/(\operatorname{null} \varphi)$ equals the dimension of the image, which is $\dim \mathbf{F} = 1$.