This is a question from a Linear Algebra textbook I am studying from, I can't figure out how to even start it, any help would be much appreciated:
$\{v_{1}, ..., v_{r}\}$ is a system of vectors in $V$ such that there exists a system of linear functionals $\{v_{1}^{v},...,v_{r}^{v}\}$ defined by :
$$ v_{i}^{v}(v_{j})=\delta_{ij} = \begin{cases} 1, & \text{if } i=j,\\ 0, & \text{if } i\neq j. \end{cases} $$
Show that if $\{v_{1}, ..., v_{r}\}$ are not generating then the dual / biorthogonal system $\{v_{1}^{v},...,v_{r}^{v}\}$ is not unique.
Again, any help is much appreciated. I understand what the dual space and dual basis are, and I've proved that that any system of vectors with a dual system must be linearly independent (by definition of dual basis and linearity), but can't figure this one out.
Let $S=\operatorname{span}(\{v_1,v_2,\ldots,v_r\})$. Since $S\varsubsetneq V$, there is some $w\in V\setminus S$. Since the set $\{v_1,v_2,\ldots,v_r,w\}$ is linearly independent, there are linear functionals $\varphi_i^w\colon V\longrightarrow k$ (where $k$ is the field that you are working with) such that $\varphi_i^w(v_j)=\delta_{ij}$ and that $\varphi_i^w(w)=1$. All these $\varphi_i^w$'s change if $w$ gets replaced by some $\lambda w$, for some $\lambda\in k\setminus\{0\}$. But all of them have the property that $\varphi_i^w(v_j)=\delta_{ij}$.