Direct sum of nontrivial vector bundles?

Question

When is it true that the direct sum (or whitney sum) of two nontrivial vector bundles is nontrivial? Also, if you have a direct sum of vector bundles, with $a$ and $b$ global sections respectively, does its direct sum have $a+b$ sections?

Answer 1

If you know the characteristic classes of your vector bundles, you can compute the characteristic classes of their direct sum, so if any of those don't vanish then the direct sum is nontrivial. Otherwise things are hard. In fact it's known that for any (real or complex) vector bundle $V$ on, say, a compact Hausdorff space, there is another vector bundle $W$ such that $V \oplus W$ is trivial.

Your second question is straightforwardly yes (assuming by "$n$ global sections" you mean "$n$ linearly independent global sections") and the most obvious construction works.