# How do you prove $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \leq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2)$? [duplicate]

Possible Duplicate:
Lower bound involving the rank of the composition of linear transformations

The following question is about a lower bound on the rank of a composition of functions given that was orgionally stated incorectly in this post Lower bound involving the rank of the composition of linear transformations.

Consider finite-dimensional vector spaces $V_1,V_2, V_3,V_4$ and linear transformations of these spaces $f_1 : V_1 \rightarrow V_2$, $f_2: V_2 \rightarrow V_3$, $f_3: V_3 \rightarrow V_4$.

How do you prove $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \leq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2)$?

-

## marked as duplicate by Did, Listing, Davide Giraudo, Mike Spivey, t.b.Nov 7 '11 at 5:14

The restriction of $f_3$ followed by the projection induces a linear map $$im(f_2)/im(f_2\circ f_1) \to im(f_3\circ f_2)/im(f_3\circ f_2\circ f_1)$$ (where $im$ is the image subspace) which is evidently surjective. The result follows.