Linear algebra helps to introduce ellipses and hyberbolas. For example an ellipse can be seen as a transformed circle by a linear application.
There is also this theorem for the curve $ax^2+2bxy+cy^2=1$ : let $t_1$ and $t_2$ denote the eigenvalues of the associated matrix.
- If $t_1>0$ and $t_2>0$, then the curve is an ellipse.
- If $t_1$ and $t_2$ have opposite signs, then the curve is a hyperbola.
Is there any interesting stuff about parabolae in linear algebra ? Sorry if my question is a bit vague.