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$\DeclareMathOperator{\Hom}{Hom}$If $T_{1}$, $T_{2} \in \Hom(V,W),$ then how could I show that $$|r(T_1)-r(T_2)|\leq r(T_1+T_2)\leq r(T_1)+r(T_2),$$ where $r(A)$ denotes the rank of $A \in \Hom(V,W)$?

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$ \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\Hom}{Hom} $Let $\Ran A$ denote the range of $A \in \Hom(V,W)$. Then:

  1. $\Ran (T_1 + T_2)$ is a priori a subspace of $\Ran(T_1) + \Ran(T_2)$, so that $$ r(T_1 + T_2) = \dim\Ran(T_1+T_2) \leq \dim\left(\Ran(T_1) + \Ran(T_2)\right) \leq \; ?$$
  2. Observe that $\Ran(T_1) = \Ran(-T_1)$ and $\Ran(T_2) = \Ran(-T_2)$. Can you apply your conclusion from 1. to $T_1 = (T_1+T_2) + (-T_2)$ and $T_2 = (T_1+T_2) + (-T_1)$?
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