= Solution
A real rank-$d$ <vector bundle> $p:E\to M$ is a continuous family of $d$-dimensional real vector spaces, locally trivial by homeomorphisms $p^{-1}(U)\cong U\times\mathbb R^d$ which are linear on every fiber. Transition matrices are continuous maps into $\operatorname{GL}_d(\mathbb R)$. The <Grassmannian> $\operatorname{Gr}_k(\mathbb R^n)$ is the space of $k$-dimensional linear subspaces; its topology can be defined by identifying each plane $W$ with its orthogonal projection matrix $P_W$. Local graph charts identify nearby planes with graphs of maps $W_0\to W_0^\perp$.
The <tautological bundle> is $E_{\rm taut}=\{(W,v):v\in W\}\to\operatorname{Gr}_k(\mathbb R^n)$. In graph charts a basis of $W_0$ gives a continuously varying basis of each graph, proving local triviality. The <universal quotient bundle> has fiber $Q_W=\mathbb R^n/W$. The map $[v]\mapsto(I-P_W)v$ identifies it continuously with the orthogonal-complement bundle $W^\perp$, whose graph charts similarly supply local frames. Thus $Q_{\rm taut}$ is a rank-$(n-k)$ <quotient vector bundle>.
For the given $E\to M$, at each $m$ choose a basis of $E_m$ and global <sections of a vector bundle> realizing its basis vectors, using the evaluation-surjectivity assumption. Those $d$ sections remain independent on a neighbourhood of $m$, since the corresponding determinant in a local trivialization is nonzero at $m$ and continuous. Compactness gives a finite cover by such neighbourhoods. Collect all the selected sections as $s_1,\ldots,s_N$; they span every fiber. They define a fiberwise surjective <vector bundle morphism>
$$
F:M\times\mathbb R^N\longrightarrow E,\qquad F_m(c_1,\ldots,c_N)=\sum_{j=1}^N c_js_j(m).
$$
Its kernels have constant dimension $N-d$ and form the <kernel bundle of a surjective vector bundle morphism>. More concretely, in a local frame of $E$ the map has a continuous full-row-rank matrix $A(m)$ and its kernel projection is
$$
P_{\ker F_m}=I_N-A(m)^T\bigl(A(m)A(m)^T\bigr)^{-1}A(m).
$$
This varies continuously and is unchanged by an invertible change of target frame. The <Grassmannian as projection matrices> therefore makes
$$
\phi(m)=\ker F_m\in\operatorname{Gr}_{N-d}(\mathbb R^N)
$$
a continuous map. The fiber quotient map induced by $F_m$ is an isomorphism $\mathbb R^N/\ker F_m\to E_m$. The same local matrix formula gives continuously varying right inverses, so these fiber isomorphisms give a <vector bundle isomorphism>. This proves the <classification of vector bundles by a universal quotient bundle>:
$$
\boxed{E\cong\phi^*Q_{\rm taut}.}
$$
The displayed equality in the classification statement is understood as bundle isomorphism, not literal equality of total spaces.
Finally, the <universal quotient bundle need not have a nowhere-zero section>. Take $\operatorname{Gr}_1(\mathbb R^2)=\mathbb{RP}^1$. Its quotient identifies with $W^\perp$, a <Möbius line bundle>. Write $W_\theta=\operatorname{span}(\cos\theta,\sin\theta)$, with $\theta$ modulo $\pi$. A section of the complement line has the form $a(\theta)(-\sin\theta,\cos\theta)$, and being single-valued requires $a(\theta+\pi)=-a(\theta)$. If $a$ is nonzero anywhere, the intermediate value theorem forces a zero between that point and its translate by $\pi$; if it is nowhere nonzero the conclusion is immediate. Hence \b[this rank-one quotient bundle has no nowhere-zero section].
Back to article page
