A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_{2n} & \dots & \alpha_{2n}^{n} \end{array}\right)$ has full rank $n+1$, provided $\alpha_{i}\neq \alpha_{j}$ for al $i\neq j$.
Can we say that a matrix of the form: $\left(\begin{array}{cccccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} & v_{0} & v_{0}\alpha_{0} & \dots & v_{0}\alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} & v_{1} & v_{1} \alpha_{1} & \dots & v_{1}\alpha_{1}^{n} \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & \alpha_{2n} & \dots & \alpha_{2n}^{n} & v_{2n} & v_{2n} \alpha_{2n} & \dots & v_{2n}\alpha_{2n}^{n} \end{array}\right)$
where $v_{0}, \dots, v_{2n}$ are constants, $v_{i}\neq v_{j}$.
will continue to have rank $(2n+1)$ ? Do any more constrains on $\{v_{i}\}$ need to be assumed ?
I know that if we consider the above matrix as $(V | M)$ where $V$ is the initial Vandermonde matrix, both $V$ and $M$ will have rank $(2n+1)$.
Edit: Sorry, I meant row rank, in all cases. The context is that I want to use the matrix $(V|M)$ to describe a solution for $2n+1$ variables $\{x_{0}, \dots, x_{2n}\}$. If this variable vector is $\vec{x}$, then I want to solve for $\vec{x}$, in: $(V|M)\cdot \vec{x} = 0$.
Hence I wanted to know whether the rows of $(V|M)$ specify linearly independent constraints on $\{x_{i}\}$.