Idea: What is ?

Is the operation mapping f to f''+3f' linear?

I know differentiation is linear, but does it hold for two separate derivatives added together?(Where f is over the collection of all infinitely differentiable functions)

Fine, A(cx + dy) = c(Ax)+d(Ay) if and only if the function is linear, but how does one actually go about picking these x and y s, or creating a matrix A to test one's theory?

For both these questions, you dont need matrices... Check if f and g are mapped as above, is f+g mapped like above....Is a function mapping f to its second derivative linear?Aye, I'm presuming this is linear, since differentiation is linear (although I'm not sure how to explain that...)

Yes it is. Rigorously you say:Is the function mapping a matrix A to its trace linear?This one is linear, right? Because you're just adding two entries together. Again, I'm not sure how to construct the function as a matrix...

Yes. Show that 0 does not map to 0, thus it is not linearIs the function mapping x to 3x + 2 linear?This is a straight out 'no', right? 3x is fine, but you can't just add a constant like that, right?

Let C = [[1,2],[3,4]]. Is the function mapping A to AC-CA linear?Again, I'm guessing it's not, for similar reasons to the previous...

This question is not clear