Linear Transformations: Proving Rules & Demonstration

[Ereisorhet]

Good afternoon people.
So i have to demonstrate that the problems below are Linear Transformations, i have searched and i know i have to do it using a couple of "rules", it is a linear transformation if:
T(u+v) = T(u) + T(v) and T(Lu) = LT(u), the thing is that i really can't understand how to develop that and find the demonstration.
Thanks for reading.

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[HallsofIvy]

For (a), let u= \begin{pmatrix}u_1 \\ u_2 \\ u_3\end{pmatrix} and u= \begin{pmatrix}v_1 \\ v_2 \\ v_3\end{pmatrix}.

We are told that "T\begin{pmatrix}x \\ y \\ z \end{pmatrix}= \begin{pmatrix}1 \\ z \end{pmatrix} so Tu= \begin{pmatrix}1 \\ u_3\end{pmatrix} and Tv= \begin{pmatrix}1 \\ v_3\end{pmatrix}. So Tu+ Tv= \begin{pmatrix}2 \\ u_3+ v_3 \end{pmatrix}.
But u+ v= \begin{pmatrix}u_1+ v_1 \\ u_2+ v_2 \\ u_3+ v_3 \end{pmatrix} so T(u+ v)= \begin{pmatrix}1 \\ u_3+ v_3\end{pmatrix}. Is T(u+ v)= Tu+ Tv?

For (b), let u= \begin{pmatrix}u_1 \\ u_2 \end{pmatrix} and v= \begin{pmatrix}v_1 \\ v_2 \end{pmatrix}. Tu= \begin{pmatrix} 2u_1+ u_2 \\ u_1- 3u_2 \\ u_1 \\ u_ 2 \end{pmatrix} and Tv= \begin{pmatrix} 2v_1+ v_2 \\ v_1- 3v_2 \\ v_1 \\ v_2 \end{pmatrix}. So Tu+ Tv= \begin{pmatrix}2u_1+ u_2+ 2v_1+ v_2 \\ u_1- 3u_2+ v_2- 3v_2 \\ u_1+ v_2 \\ u_2+ v_2 \end{pmatrix}.
u+ v= \begin{pmatrix}u_1+ v_1 \\ u_2+ v_2 \end{pmatrix} so T(u+ v)= \begin{pmatrix}2(u_1+ v_1)+ (u_2+ v_2) \\ (u_1+ v_1)- 3(u_2+ v_3) \\ u_1+ v_1 \\ u_2+ v_2 \end{pmatrix}.

Do you see that those are the same, so T(u+ v)= Tu+ Tv?

Now we need to show that "T(Lu)= LTu" where L is a "scalar" (a number). Lu= \begin{pmatrix}Lu_1 \\ Lu_2 \end{pmatrix} so T(Lu)= \begin{pmatrix} 2Lu_1+ Lu_2 \\ Lu_1- 3Lu_2 \\ Lu_1 \\ Lu_2 \end{pmatrix} and LTu= L\begin{pmatrix}2u_1+ u_2 \\ u_1- 3u_2 \\ u_1 \\ u_2\end{pmatrix}= \begin{pmatrix}L(2u_1+ u_2) \\ L(u_1- 3u_2) \\ Lu_1 \\ Lu_2 \end{pmatrix}. Do you see that T(Lu)= LTu?