I have a few questions about transformations:
Any help regarding this content is greatly appreciated. I'd also love some reference material to clarify my understanding on the topic.
On
A linear transformation from a vector space $V$ to another, $W$.
Is a function $L : V \rightarrow W$ s.t. $L(x+y) = L(x)+L(y), \forall x,y \in V$.
And $L(c x) = cL(x), \forall x \in V, c \in F$. where $F$ is the base field.
So in essence, is a function that maps straight lines to straight lines.
So the first problem of your question is when you say "transformation from the circle to the sphere". It doesn't make sense to talk about linear transformation unless your domain is a vector space, by "circle" do you mean the curve or the unit disk including its boundary? Either way they do not form a vector space under the usual addition and multiplication.
so I'm going to assume you are talking about the function $f(x,y)= (x,y,\sqrt{1-x^2-y^2})$ from $\mathbb{R}^2 \rightarrow \mathbb{R}^3$ with $\mathbb{R}^2$ and $\mathbb{R}^3$ being vector spaces over $\mathbb{R}$. If so, no, this function is not linear, since $f(2x,0) \neq 2f(x,0)$.
However, the map $g(x,y,z) = (x,y)$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ indeed is a linear transformation (called projection). And $g$ does send the sphere to the unit disk.
No, there isn't a linear transformation that isn't a matrix transformation, every linear transformation has an associated matrix representation. This is because how a linear map "look" like is completely determined by where the basis elements are sent to.
Hope that answers your question.
Edit: unit disk not circle
One talks of linear transformations for vector spaces. Remember a vector space comes with two operations: addition of two vectors, and multiplication of a vector by a number (also called scalar).
A linear transformation is function between two such vector spaces for which if those two operations are carried out before applying the function or after applying the function yield the same answer:
Add the vectors and then apply the function: given $u, v$ calculate $u+v$ first and then $T(u+v)$.
Apply function first then add the vectors: given $u,v$ calculate $T(u)$ and $T(v)$ separately first. Then add those two vectors getting $T(u) + T(v)$.
The final result in the two different approaches are $T(u+v)$ and $T(u)+T(v)$; these two are the same vectors for a linear transformation.
Similarly given a number $c$ (i.e a scalar) and a vector $v$ first do scalar multiplication then apply function getting $T(c.v)$ .
Alternatively apply function first and the do scalar multiplication $ c. T(v)$. These two are equal for a linear transformation: T(c.v)= c.T(v)$
(Mixing decoction with milk and then drinking it I feel I had taken coffee. Drinking decoction and milk separately and then allowing it to mix in my stomach I don't get the same effect. So drinking has a non-linear transformational effect on my body!)