Dimension of the preimage of a vector subspace ($\dim T^{-1}(X)=\dim (X\cap \mathrm{im}T)+\dim \ker T$)

Let $T:U\to V$ be a linear map between finite-dimensional vector spaces and $X$ a vector subspace of $V$. The preimage $T^{-1}(X)$ is a vector subspace containing $\ker T$. To compute its dimension, choose a basis $k_1,\dots ,k_s$ of $\ker T$ and a basis $y_1,\dots ,y_d$ of $X\cap \mathrm{im}T$. Lift each $y_j$ to $u_j\in U$ with $T{u}_j=y_j$. Then $k_1,\dots ,k_s,u_1,\dots ,u_d$ is a basis of $T^{-1}(X)$: applying $T$ proves linear independence, and subtracting a suitable linear combination of the lifts reduces every element of the preimage to the kernel. This proves the displayed formula without using the rank-nullity theorem. Taking $X=V$ then proves the rank-nullity theorem itself.