2 linear algebra questions 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. 
 A: 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 $\mathbb{R}^2$?
A: 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.
