= Direct sum of vector bundles
{title2=$(E\oplus F)_x=E_x\oplus F_x$}
For two <vector bundles> $E,F$ over the same base space $X$ and the same scalar <field>, their <direct sum of vector bundles> is a bundle with fibers
$$
(E\oplus F)_x=E_x\oplus F_x.
$$
Its total space consists of pairs $(e,f)$ lying above the same $x$, with the topology inherited from the fiber product. If $E$ and $F$ have ranks $r$ and $s$, simultaneous <local trivializations> give fiber $k^{r+s}$, and their <transition functions of a vector bundle> combine in block-diagonal matrices. Thus the ranks add. This differs from the <tensor product of vector bundles>, whose ranks multiply.<Vector Bundles and K-Theory by Allen Hatcher>[section 1.1, Direct Sums]{ref}
A geometric example comes from the unit sphere $S^2\subset\mathbb R^3$. At each $x$, tangent vectors form the plane perpendicular to $x$ and normal vectors form the line spanned by $x$. Their direct sum is all of $\mathbb R^3$, continuously in $x$. Consequently $TS^2\oplus\varepsilon^1\cong\varepsilon^3$, where $\varepsilon^j$ denotes the <trivial vector bundle> of real rank $j$. Explicitly, $(v,t)\mapsto v+tx$ is the fiberwise isomorphism. Adding one trivial direction can therefore trivialize a nontrivial <tangent bundle>; the <Hairy ball theorem> prevents $TS^2$ itself from being trivial.
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