Hint for proof of proposition 10.15, lines a) and c) in John M. Lee's book "An Introduction to Smooth Manifolds"

Question

I have been thinking about how to prove proposition 10.15, lines a) and c) in John Lee's book "An Introduction to Smooth Manifolds" (I leave a screenshot of the proposition below), but I can't figure out how to construct the remaining sections in a). Each section can be written locally as a linear combination of the local canonical smooth frame $t_{i}: V \rightarrow E$ such that, using a local trivialization $\Phi : \pi ^{-1}(V) \to V\times \mathbb{R}^k$, we get $\Phi\circ t_{i} : V \to V \times \mathbb{R}^k$ is given by $\Phi \circ t_{i} (p) = (p,e_{i})$. So, $\sigma _{j} = \sum_{i=1}^{k} \lambda_{i,j} t_{i}$ for $1\leq j \leq m \lt k$, where the $\lambda_{i,j}$'s are smooth as real valued functions on $V$. So the statement that we can extend the linearly independent set $\{\sigma_{j}\}_{j=1,...,m}$ to a local frame is equivalent to stating that the matrix valued function $[\lambda_{i,j}]_{i = 1,...,k}^{j = 1,...,m}$, which is a matrix with maximal rank at each point, can be "extended" to a smooth map $M: W \subset V \to GL(k,\mathbb{R})$.

Having said this, I would appreciate it if you could give me a hint for the idea of the proofs of a) and c) :).

Thank you all in advance!

Proposition 10.15 (Completion of Local Frame for Vector Bundles). Suppose $\pi:E\longrightarrow M$ is a smooth vector bundle of rank $k$.

(a) If $(\sigma_1,\ldots,\sigma_m)$ is a linearly independent $m$-tuple of smooth local sections of $E$ over an open subset $U\subseteq M$, with $1\leq m<k$, then for each $p\in U$ there exist smooth sections $\sigma_{m+1},\ldots,\sigma_k$ defined on some neighborhood $V$ of $p$ such that $(\sigma_1,\ldots,\sigma_k)$ is a smooth local frame for $E$ over $U\cap V$.

(b) If $(v_1,\ldots,v_m)$ is a linearly independent $m$-tuple of elements of $E_p$ for some $p\in M$, with $1\leq m<k$, then there exists a smooth local frame $(\sigma_i)$ for $E$ over some neighborhood of $p$ such that $\sigma_i(p)=v_i$ for $i =1,\ldots,m$.

(c) If $A\subseteq M$ is a closed subset and $(\tau_1,\ldots,\tau_k)$ is a linearly independent $k$-tuple of sections of $E|_A$ that are smooth in the sense described in Lemma 10.12, then there exists a smooth local frame $(\sigma_1,\ldots,\sigma_k)$ for $E$ over some neighborhood of $A$ such that $\sigma_i|_A=\tau_i$ for $i=1,\ldots,k$.

Exercise 10.16. Prove the preceding proposition.

Answer 1

You almost got it. Continuing what you said, at point $p$ we can extend the matrix $$\begin{bmatrix} \lambda_1^1(p) & \cdots & \lambda_m^1(p)\\ \vdots & \vdots& \vdots\\ \lambda_1^k(p) & \cdots & \lambda_m^k(p) \end{bmatrix}$$ to a $GL_k(\mathbb{R})$ matrix by extending the linear independent vectors $$v_1(p):=\begin{bmatrix} \lambda_1^1(p) \\ \vdots \\ \lambda_1^k(p) \end{bmatrix},\dots,v_m(p):=\begin{bmatrix} \lambda_m^1(p) \\ \vdots \\ \lambda_m^k(p) \end{bmatrix}$$ to a basis of $\mathbb{R}^n$. Let $v_1(p),\dots v_m(p),v_{m+1},\dots,v_k$ be that basis. You have to show that the map $F:U \to \mathbb{R}$ given by $$p \mapsto \det\left(\begin{bmatrix} v_1(p) & \cdots & v_m(p)& v_{m+1} & \cdots & v_k \end{bmatrix}\right)$$ is smooth. Use the fact that $F^{-1}(0)$ is closed to conclude.