# [Math] Verify if a function is a linear transformation (R3 to R3)

linear algebralinear-transformations

I have an algebra final exam on tuesday, and I'm having some trouble with this kind of exercise. I tried looking for an explanation in all over the web, but couldn't find anything of this type. Of course, I also searched a lot here, but really didn't find what I need.

Ok, so: I know that, for a function to be a linear transformation, it needs to verify two properties:

1: T (u+v) = T (u) + T (v)

2: c.T (u) = T (c.u)

This is what I will need to solve in the exam, I mean, this kind of exercise:

T: R3 -> R3 / T (x; y; z) = (x+z; -2x+y+z; -3y)

The thing is, that I can't seem to find a way to verify the first property. I'm writing nonsense things or trying to do things without actually knowing what I am doing, or why I am doing it. I first tried to make two generic vectors, then tried to apply the function on it… but it looks horrible. Fun fact: I'm not having big troubles with solving R2 to R2 exercises, but I can't seem to find a way to solve this one.

How should I start to verify the first property in this kind of exercise? Note that in this case we shouldn't use matrix or anything, it should be just pure math.

Let $$u=(x_1,y_1,z_1)$$ and $$v=(x_2,y_2,z_2)$$.
Can you then show $$T(u+v)=T(u)+T(v)$$?