Question
For the matrix $$ A=\begin{pmatrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \\ \end{pmatrix} $$ the set $$S=\{\textbf b \in \mathbb{R^3} : \textbf{b} = A\textbf{x} \text{ for some }\textbf x \in \mathbb{R^3}\}$$ is the column space for $A$.
Find the condition for augmented matrix $A\textbf{v}$ so that the vector $ \textbf{v}=\begin{pmatrix} x \\ y \\ z \\\end{pmatrix} $ belongs to the set $S$.
My Attempt
We find that saying $\textbf v \in S$ is equivalent to $x,y$ and $z$ satisfying a linear condition of the form $ax+by+cz=0$. Specifically, this condition is equivalent to?
I'm trying to find the linear condition by row reducing and Gaussian Elimination method, however this is the first time I've seen an augmented matrix where the $x, y$ and $z$ is on the right hand side of the vertical line like in the question.
Does this mean I can make these assumptions?
$a-2b+2c = x\\ a-4b+2c = y\\ -6a+18b-12c = z$Do I then find the value of $a, b,$ and $c$ in terms of $x, y$ and $z$ and substitute it to $ax+by+cz=0$ ?