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- Jan 26, 2012

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**Problem**: Let $U$, $V$, and $W$ be three left $K$-vector spaces, and $\psi$, $\phi$ linear maps, fitting into a short exact sequence:

\[ 0\rightarrow U \xrightarrow{\psi} V \xrightarrow{\phi} W \rightarrow 0.\]

Define

\[S = \{\sigma \in \text{Hom}_K(W,V) : \phi \circ \sigma = \text{Id}_W\}.\]

(An element of S is called a

*splitting*of the short exact sequence). Prove that there exists a bijection from $\text{Hom}_K(W,U)$ to $S$.

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