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Let $f,g: V \rightarrow W$ be linear operators. Prove that $r(f + g) \leq r(f) + f(g)$

Note: r = rank

My idea was to use matrix representation and to prove that the rank of matrix C ($C= F+G$) can't be bigger than the sum of $r(F) + r(G)$,but to use that idea I must prove that $r(f) = r(g)$ which I don't have an idea how to do. Other idea was to prove that $(f+g)(v) = f(v) + g(v)$ but also with no luck.

Dragan Zrilić
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    You have a theorem that says $\ r(A+V)=r(A)+r(V)-r(A\cap{V})$ – CTSnake Apr 25 '17 at 21:40
  • This question has been asked here many times before and you should do a search. – AnyAD Apr 25 '17 at 22:01
  • Any linear operator can be represented by a matrix, so the question reduces to one concerning matrices. – AnyAD Apr 25 '17 at 22:03
  • Note the ranks of $f $ and $g $ don't need to be equal and similarly you will not necessarily get the equality in the expression. – AnyAD Apr 25 '17 at 22:24

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Hint: $\operatorname{im}(f+g)\subseteq \operatorname{im}f+\operatorname{im}g$

Ennar
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