Applying linear transformations to equations of curves I am currently learning linear algebra. I wonder if it possible to use linear transformations to find equations of curves that are translated or rotated. For example how can I use matrices to find an equation of a parabola rotated by 45 degrees clockwise?
 A: The calculus is
$$\left(\begin{array}{cc}\cos(-\pi/4)&-\sin(-\pi/4)\\\sin(-\pi/4)&\cos(-\pi/4)
\end{array}\right)\left(\begin{array}{c}x\\x^2\end{array}\right).$$
That is because the matrices of the form
$$\left(\begin{array}{cc}\cos(\theta)&-\sin(\theta)\\\sin(\theta)&\cos(\theta)
\end{array}\right)$$ rotate the points on the plane by a $\theta$ angle.
A: A linear transformation $L$ sends a point $\vec x$ to $\vec x'=L(\vec x)$. You want to apply this to all points on some curve. (It also works for surfaces etc.)
If you have a parametric equation $\vec x=f(t)$, then you can apply the transformation to the output: $\vec x'=L(\vec x)=L(f(t))$. In other words the new parametric function is $L\circ f$.
If you have an implicit equation $f(\vec x)=0$, then you can apply the inverse transformation to the input: $f(L^{-1}(\vec x'))=f(\vec x)=0$. In other words the new implicit function is $f\circ L^{-1}$.
(The standard parabola has parametric equation $(x_1,x_2)=(t,t^2)$ and implicit equation $x_1^2-x_2=0$.)
