Question

Let f : R → R3 be defined by f(x) = <4x,−3x,5x+5>. Is f a linear transformation

Answer 1

since each component is linear, I'd say yes.

Answer 2

of course, to prove it, you'd need to show that
f(cx) = c*f(x)
f(x+y) = f(x)+f(y)
...
huh. I guess not linear.