Naive linear algebra in vector bundles

Question

I would like to ask a fairly general question: how much of naïve linear algebra remains true for vector bundles? For example,

1. If $F$ and $F’$ are two subbundles of $E$ such that the fiber $E_x$ is the internal direct sum $F_x$ and $F’_x$ at each point $x$, is it true that $E$ is isomorphic to the direct sum of $F$ and $F’$?

2. Is the quotient of $E\oplus F$ by $E$ isomorphic to $F$?

3. Is the direct sum $E\oplus F$ the product / coproduct in the category of vector bundles over a fixed manifold? What if the base is not fixed?

My intuition says that the answers to these questions should be yes, but intuition for linear algebra could fail for vector bundles. If they are true, could you provide a reference for the proof or at least a general idea of the proof?

Answer 1

Question 3. is already addressed in Federico T.'s answer. Here is one way of thinking about Questions 1. and 2. that I like. The most important fact here is, in my opinion, the

Proposition: If $E,F$ are vector bundles over a topological space $X$ and $f\colon E\rightarrow F$ is a bundle map, then $f$ is a bundle isomorphism if and only if it is a fiber-wise linear isomorphism.

Together with universal properties, this answers your questions. For question 1., the subbundles come with inclusion bundle maps $F\rightarrow E$ and $F^{\prime}\rightarrow E$, which induce a bundle map $F\oplus F^{\prime}\rightarrow E$. This is a bundle isomorphism iff it is a fiber-wise linear isomorphism iff, for every point $x$, $E_x$ is the internal direct sum of $F_x$ and $F^{\prime}_x$. Question 2. can be addressed similarly.

Answer 2

Basically, quite a lot! And the reason is quite simple: as the vector bundles are locally trivial, you can use a local frame to repeat most of the proofs you do in linear algebra using a basis. In particular the facts you guessed are true and the proofs are not different from the ones you would do for vector spaces. In Milnor-Stasheff "Characteristic Classes" you can find the proofs or some hints to check them.

In the category of vector bundles over a fixed space, the Whitney sum is a biproduct, and kernel/cokernel exist for all morphism of constant rank. This makes it an additive category.

If you let change the base space, things go slightly worse: if $E \to M$ and $F \to N$ are vector bundles, their Whitney sum is obtained summing their pullbacks along the projections $$M \leftarrow M \times N \to N.$$ Therefore, their sum $E \oplus F$ is still a vector bundle, but now over $M \times N$. This still behaves as a product in this category, but it is no more a coproduct as there is no canonical way to inject the factors into their product.

A nice remark is that linear algebra can be recovered from the theory of vector bundles just observing that any vector space is a vector bundle over a point. This lets you embed it in this context in a very elegant way.