I'm trying to find an example of a function from $\mathbb{R}^2 \to \mathbb{R}^2$ which is not linear. So would this be something like a parabola or more like a vertical line?
Also, I'm trying to find a line, $L$, where $f(L)$ is not a straight line. This one has me stumped. I'm supposed to prove it and I'm not sure I can even think of one?
I don't intend for people to solve the problems for me but some advice on where to start or how to think and formulate an answer is much appreciated.
This is not answer as requested. It is a explanation that can hopefully give you a more intuitive understanding to help you come up with your proofs.
A linear function is any function that satisfies two properties: $ \gamma f(\vec x)= f(\gamma \vec x) $ and $ f(\vec x + \vec y)= f( \vec x)+ f(\vec y)$ where $\vec x \vec y $ are vectors and $\gamma $ is a scalar.
You're instinct to think of a parabola is correct. What you want is a function that doesn't satisfy the two properties. Basically a linear function is a function that geometrically speaking can stretch and change directions of vectors but does not transform a straight line into a any graph with a curvature. This has to do with the fact that the first property $ \gamma f(\vec x)= f(\gamma \vec x) $ only stretches or contracts lines and by the second property $ f(\vec x + \vec y)= f( \vec x)+ f(\vec y)$ that all points in a vector space remain proportional distance apart after a linear transformation.
I highly recommend checking out this video that uses very elegant computer graphics to make this point intuitively which talks about the the geometric movement of linear functions:
https://www.youtube.com/watch?v=kYB8IZa5AuE&index=4&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab
** Also a linear function always maps a $\vec 0 $ to $\vec 0$ which is why JMoravitz solution works.
For your second question it is basically the same as the first. Hint $f(x)= e^x$ which maps the x-axis, a straight line, to the curve of $e^x$. Now think of this with context of $\mathbb R^2 \rightarrow \mathbb R^2$.
Hope this helps.
Hint for the first question: Can you show that $f(x)=x+1$ is not a linear function from $\mathbb{R}$ to $\mathbb{R}$? Can you generalize this to a function from $\mathbb{R}^2$ to $\mathbb{R}^2$?
Hint for the second question: Can you find a function from $\mathbb{R}$ to $\mathbb{R}$ that sends the real line to a point? Can you generalize this result to a function from $\mathbb{R}^2$ to $\mathbb{R}^2$?