Bundle that is isomorphic to the bundle of Whitney sum

Question

I am involved with one question that a friend of mine asked.

If a vector bundle $(E,\pi,M)$ has two sub-bundles, $(F,\xi,M)$, $(L,\psi,M)$, and $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ then is true that $E$ is isomorphic to the Whitney sum of $F$ and $L$?

It is quite obvious that is true, but I have no expertise on a construction on morphism between bundles, could some one give a hint about how this isomorphism can be written?

Thanks!

Answer 1

Let $i_F:F\rightarrow E$ and $i_L:L\rightarrow E$ be the canonical embeddings. Let $F+L$ the Whitney sum of $F$ and $L$. Denote by $j_F:F\rightarrow F+L$ and $j_L:L\rightarrow F+L$ the canonical embeddings.

The universal property of the sum defines a morphism $f:F+L\rightarrow E$ such that $j_F\circ f=i_F$ and $j_L\circ f=i_L$ which is in fact an isomorphism.