# Thread: A proof regarding rank

1. ## A proof regarding rank

$\displaystyle rank(\alpha+\beta) \leq rank(\alpha) + rank(\beta)$

2. Originally Posted by szpengchao
$\displaystyle rank(\alpha+\beta) \leq rank(\alpha) + rank(\beta)$
'rank' is a very broad word and has many different meanings in different topics in mathematics.

What topics are you talking about and what do alpha and beta represent?

3. ## linear algebra

they r linear maps from V to W.

4. So the "rank" of $\displaystyle \alpha$ is the dimension of $\displaystyle \alpha(V)$ as a subspace of W. If w is in $\displaystyle (\alpha+ \beta)(V)$ then there exist v in V such that $\displaystyle (\alpha+ \beta)v= \alpha(v)+ \beta(v)$ is in W. Certainly that is true if $\displaystyle \alpha(v)$ and $\displaystyle \alpha(v)$ are in W. Do you see why that means that $\displaystyle (\alpha+ \beta)V$ must be a subspace of the direct sum $\displaystyle \alpha(V)$ and $\displaystyle \beta(V)$?

5. ## ok.

$\displaystyle Im(\alpha+\beta)\subset Im(\alpha)\bigoplus Im(\beta)$
$\displaystyle \forall w\in Im(\alpha+\beta), \exists \ \ unique \ \ v\in V, \ \ s.t. \ \ (\alpha+\beta)(v)=w \ \ \Rightarrow w\in Im(\alpha)\bigoplus Im(\beta)$