Rank-nullity theorem and dimension formula for subspaces

Question

Let $V,W$ be vector spaces, $U_1, U_2 \subseteq V$ subspaces and $\phi \in Hom(V,W)$. Then how can it be shown that $dim (U_1+U_2)=dim(U_1)+dim(U_2)-dim(U_1 \cap U_2)$, and $dim V = dim \ im \ \phi + dim \ ker \ \phi$ are equivalent? I've only seen a proof where it is used that $dim (V / W) = dim \ V - dim \ W$. How can the equivalence be shown directly?

Answer 1

First of all, presumably the proof you're referring to uses the quotient space $V/W$.

To answer your question, one proof is as follows: begin by constructing a basis $\{v_1,\dots,v_d\}$ of $U_1 \cap U_2$. Let $m = \dim U_1$ and $n = \dim U_2$. This basis above can be extended to bases $$ B_1 = \{v_1,\dots,v_d,v_{d+1}^{(1)},\dots,v_m^{(1)}\}\\ B_2 = \{v_1,\dots,v_d,v_{d+1}^{(2)},\dots,v_n^{(2)}\} $$ of the spaces $U_1$ and $U_2$ respectively. It suffices to show that the set $$ \{v_1,\dots,v_d,v_{d+1}^{(1)},\dots,v_m^{(1)},v_{d+1}^{(2)}\dots,v_n^{(2)}\} $$ is linearly independent and spans $U_1 + U_2$.