From S.L linear algebra:
Let $F:R^2 \rightarrow R^2$ be the mapping given by $F(x, y)=(2x, 2y)$. Describe the image under $F$ of the points lying on the circle $x^2+y^2=1$.
Let $(x, y)$ be a point on the circle of radius $1$. Let $u=2x$ and $v=2y$. Then $u$, $v$ satisfy the relation $(u/2)^2+(v/2)^2=\frac{u^2}{4}+\frac{v^2}{4}=1$. Hence $(u, v)$ is a point on the circle of radius $2$. Therefore the image under $F$ of the circle of radius $1$ is a subset of the circle of radius $2$.
Conversely, given a point $(u, v)$ such that $u^2+v^2=4$, Let $x=\frac{u}{2}$ and $y=\frac{v}{2}$. Then the point $(u, v)$ satisfies the equation $x^2+y^2=1$, and hence is a point on the circle of radius $1$. Furthermore, $F(x, y)=(u, v)$. Hence every point on the circle of radius $2$ is the image of some point on the circle of radius $1$. We conclude finally that the image of the circle of radius $1$ under $F$ is precisely the circle of radius $2$.
The text above is an example of how an image of certain subset can be described. In the case above image of all points of the circle under $F$ are described in slightly implicit manner, but is still understandable.
There is an exercise:
Let $F:R^2 \rightarrow R^2$ be the mapping defined by $F(x, y)=(2x, > 3y)$. Describe the image of the points lying on the circle $x^2+y^2=1$.
If I'm correct, an image under $F$ in this case would be an ellipse with input dependent major/minor axises.
Following the logic in the text above, we would probably have:
Let $(x, y)$ be a point on the circle of radius $1$. Let $u=2x$ and $v=3y$, then u,v satisfy the relation $(u/2)^2+(v/3)^2=\frac{u^2}{4}+\frac{v^2}{9}=1$. Therefore $u$ is a point on the circle of radius $2$ and $v$ is a point on the circle of radius $3$. Hence the image under $F$ of circle of radius $1$ is the circle of radius $2$, and the image under $F$ of circle of radius 2 is the circle of radius $3$.
But that seems rather confusing. Hence, what would be the most explicit and proper way of describing sets under images of any arbitrary $F$?
Thank you!
The image of the unit circle by a linear transformation of the plane is an ellipse. The directions and lengths of its axes can be found by computing the singular value decomposition of the transformation.
(image from Wikipedia)