Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$\DeclareMathOperator{\Hom}{Hom}$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 $A \in \Hom(V,W)$?

share|improve this question

1 Answer 1

$ \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 $T_2 = (T_1+T_2) + (-T_1)$?
share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.