Let $C[a,b]$ be the set of continuous functions on $[a,b]$, then a linear subspace $V \subseteq C[a,b]$ of finite dimension $n+1$ is called an Haar subspace iff one of the following equivalent conditions hold:
1) every non-zero element $p \in V$ has at most $n$ zeros,
2) for $n+1$ pairs $(t_i, f_i)$ there exists exactly one $p \in V$ interpolating between those values
3) is $h_0, \ldots, h_n$ any base for $V$, and are $t_0, \ldots, t_n \in [a,b]$ distinct point, we have $$ \det \begin{pmatrix} h_0(t_0) & \cdots & h_n(t_0) \\ \vdots & & \vdots \\ h_n(t_n) & & h_n(t_n) \end{pmatrix} \ne 0. $$ Now I want to show that the set $\{ 1, x, xe^{2x} \}$ is a Haar subspace of $C[0,1]$. But the difficulty arises with the non-linearity of the involved equations, which I cannot solve. So do you have any ideas how to prove this?
Let's write $f(x) = xe^{2x}$. Let $p $ be a nonzero linear combination of $1,x,f(x) $. If $p$ does not involve $f$, then it's linear. Suppose it does. Then the zeros of $p$ correspond to intersections of the graph of $f$ with some line.
Is there a line that meets the graph of $f$ more than twice? Spoiler below.