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I need to show a given function is a linear transformation. I know it needs to hold under addition and multiplication of scalars. I struggle with knowing where to start. For example, I have a problem that says: $L\colon \mathbb R^3\to \mathbb R^2$ defined by $L(x,y,z)=(x,z)$.

Would I start with $x_1 + x_2$? Or $x_1 + z_1$? (I now understand that I should begin with x+x' and so on)

I am asking for clarification on the formatting such as: L(x, y, z)= L(x+x', y+y', z+z') From this step should I state the transformation: L(x+x', y+y', z+z') = L (x,z) = L(x+x', z+z')?

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Following your example, $L(x,y,z)=(x,z)$, you will have to prove that

  1. $L((x,y,z)+(x',y',z'))=L(x,y,z) + L(x',y',z')$
  2. $L(\lambda\cdot(x,y,z))=\lambda L(x,y,z)$

Let us do (2) as an example:

$$L(\lambda\cdot(x,y,z))=L(\lambda x, \lambda y, \lambda z)=(\lambda x, \lambda z)=\lambda \cdot (x,z)=\lambda \cdot L(x,y,z)$$

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    Once you started (2) I fully understood where you were going so let me see if I can get (1). edit: So would this be as simple as saying L(x+x',y+y',z+z')= L(x,y,z)+(x',y',z')? – 2017-02-10
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    I guess you wanted to say $L(x+x',y+y',z+z')= L(x,y,z)+L(x',y',z')$. And yes, that would be the case. – 2017-02-10
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    @KellyR yes, this is exactly what does "holding under addition" means. – 2017-02-10
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    Thanks! I know what I need to show I just struggle with knowing how to show it. If I can figure out what I should start with, like here I wasn't sure if I should do L(x+x', y+y', z+z') = L(x, y, z) + L(x, y', z'') = L (x, z) + L(x', z')? I guess my original question should have stated more of I'm not sure about formatting. (I'll edit my original question) – 2017-02-10
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    What is $L(x,z)$ at all? Just use the definitions of $L$ and addition in $\mathbb R^2$ – 2017-02-10
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    L(x,z) is our transformed function isn't it? Don't I need to show that when proving it's a transformation? – 2017-02-10
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    $L(x,z)$ has no sense. What you have is $L(x,y,z)$ and that is equal to $(x,z)$. Think of $L$ as "forgetting about the $y$". Given an element of $\mathbb R^3$, $(x,y,z)$, $L(x,y,z)$ is the result of taking the $x$ and the $z$ coordinates. – 2017-02-10