How to understand the result of Mathematica's CoefficientList?
Matrix of coefficients for a quadratic form:
CoefficientList[1 + a x^2 + b x y + c y^2, {x, y}]
$$\begin{align*}\left(\begin{array}{ccc} 1 & 0 & c \\ 0 & b & 0 \\ a & 0 &
0 \\\end{array}\right)\end{align*}$$
why this represent a quadratic form?
CoefficientList[a x^2 + b x y + c y^2 + d x + e y + f, {x, y}]
$$\begin{align*}\left(\begin{array}{ccc} f & e & c \\ d & b & 0 \\ a & 0 &
0 \\\end{array}\right)\end{align*}$$
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