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## Homework Statement

For the following mappings, state the domain and the codomain. Determine whether the mapping is linear by using the definition of linearity: either prove it is linear or give a counterexample to show why it cannot be linear.

i.) f(x1,x2,x3)=(2x2, x1−x3)

ii.) g(x1, x2) = (cos x2, x1x2^3)

iii.) h(x1, x2, x3) = (0, 0, x1 + x2 + x3)

I don't know how to see or interpret this notation. I've looked through my notes and can't seem to find anything that even remotely resembles this type of problem. Because I cannot understand the question, I'm having a hard time at even an attempt for a solution.

**2. The attempt at a solution**

Looking at just i. right now:

I would like to say the domain is R^3 since there are three columns on the left side... I don't really know if that is how to find the domain. Is the question kind of saying something like... f(x,y,z) = 2y, x-z ?

I do know, however, that if g(a,b,) = (ga,gb), and g(k(a,b)) = kg(a,b), then that is proof of linearity. But reading this notation, it feels way too abstract for me to wrap my head around. Any help would be greatly appreciated, thank you.

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