construction of linear independent local section in vector bundle

Question

Let $X_1,\ldots,X_k$ be lineary independent local sections of a vector bunlde $V$, $rank(V)>k$, defined on some open set $U$. Can I construct a local section $X_{k+1}$ on a (maybe smaller) open subset of $U$, such that $(X_1,\ldots,X_{k+1})$ are lineary independent on the smaller subset?

edit: I know that answer, however I would be interested in a more "elementary" argument.

Answer 1

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Basis}{\mathbf{e}}$Let $N$ denote the rank of $E$ and $n$ the dimension of the base. By working in a trivializing neighborhood for $E$, and identifying local sections of $E$ with $\Reals^{N}$-valued functions without using separate notation, the question boils down to:

Suppose $(\Basis_{i})_{i=1}^{N}$ is the standard basis of $\Reals^{N}$, $U$ is an open neighborhood of the origin in $\Reals^{n}$, and $(X_{j})_{j=1}^{k}$ are continuous functions from $U$ into $\Reals^{N}$ such that:

• $X_{j}(0) = \Basis_{j}$ for $1 \leq j \leq k$.

• $\bigl(X_{j}(p)\bigr)_{j=1}^{k}$ is an orthonormal set for each $p$ in $U$. (Perform Gram-Schmidt in $\Reals^{N}$ if necessary.)

Does there exist an open set $U' \subset U$ and a continuous function $X_{k+1}:U' \to \Reals^{N}$ such that $(X_{j})_{j=1}^{k+1}$ is pointwise linearly independent in $U'$?

The answer is clearly "yes". Consider the constant function $X_{k+1} = \Basis_{k+1}$. The orthogonal projection of $X_{k+1}$ into the space spanned by $(X_{j})_{j=1}^{k}$, and its squared magnitude, are given by $$ X_{k+1}' := \sum_{i=1}^{k} \langle X_{k+1}, X_{i}\rangle\, X_{i},\qquad \|X_{k+1}'\|^{2} = \sum_{i=1}^{k} \langle X_{k+1}, X_{i}\rangle^{2}. $$

By elementary linear algebra, $(X_{i})_{i=1}^{k+1}$ is linearly independent if and only if $X_{k+1} \not\in \operatorname{Span}(X_{i})_{i=1}^{k}$, if and only if $X_{k+1} \neq X_{k+1}'$. But the squared magnitude of $X_{k+1}'$ is $0$ at the origin, and is continuous as a real-valued function on $U$, so there exists an open set $U'$ containing the origin such that $\|X_{k+1}'\|^{2} < 1 = \|X_{k+1}\|^{2}$ on $U'$.