Converting a input-output equation to matrix form

Question

I have the following system of equations: $$y_1=a_{11}x_1+a_{21}x_2+a_{31}x_1^2+a_{41}x_1x_2$$ $$y_2=a_{12}x_1+a_{22}x_2+a_{32}x_1^2+a_{42}x_1x_2$$ $$y_3=a_{13}x_1+a_{23}x_2+a_{33}x_1^2+a_{43}x_1x_2$$ How can I convert it to matrix form: $Ax=B$ given that $x = [x_1,x_2]^T$? Do I need to use another $x$ vector, like $x = [x_1, x_2, x_1^2, x_1x_2]^T$?

Answer 1

I dont know the proper way but i imagine this would be the way to go.

$\begin{pmatrix} a_{11} & a_{21} & a_{31} & a_{41} \\ a_{12} & a_{22} & a_{32} & a_{42} \\ a_{13} & a_{23} & a_{33} & a_{43} \\ \end{pmatrix}$ $\huge.$ $\begin{pmatrix} x_1 \\ x_2 \\ x_1^2 \\ x_1x_2 \end{pmatrix}$= $\begin{pmatrix} y_1\\y_2\\y_3 \end{pmatrix}$

Where A = $\begin{pmatrix} a_{11} & a_{21} & a_{31} & a_{41} \\ a_{12} & a_{22} & a_{32} & a_{42} \\ a_{13} & a_{23} & a_{33} & a_{43} \\ \end{pmatrix}$ , X =$\begin{pmatrix} x_1 \\ x_2 \\ x_1^2 \\ x_1x_2 \end{pmatrix}$ and B = $\begin{pmatrix} y_1\\y_2\\y_3 \end{pmatrix}$ in the form that you wanted.