Linear Transformations

[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.

 

[pka]

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.

View attachment 12306

To show that \(\displaystyle T\) is a linear transformation on \(\displaystyle \mathcal{V}\) over a field \(\displaystyle \mathcal{F}\) it suffices to show \(\displaystyle (\forall u~\&~v\in\mathcal{V})(\forall\alpha\in\mathcal{F})[T(u+\alpha v)=T(u)+\alpha T(v)\)
Moreover there is a quick test to prove it is not a linear transformation: Does it map the zero to zero.
For part a), does \(\displaystyle T\left( {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right) \to \left( {\begin{array}{*{20}{c}}0\\0\end{array}} \right)~?\)
Give part b) a try and post your effort.

 

[HallsofIvy]

You do it! For (a) let \(\displaystyle u= \begin{pmatrix}x_1 \\ y_1\\ z_1\end{pmatrix}\) and \(\displaystyle v= \begin{pmatrix}x_2\\ y_2\\ z_2 \end{pmatrix}\). Then \(\displaystyle Tu= \begin{pmatrix}1 \\ z_1\end{pmatrix}\), \(\displaystyle Tv= \begin{pmatrix}z_2 \\ 1 \end{pmatrix}\) so that \(\displaystyle Tu+ Tv= \begin{pmatrix}z_1+ z_1 \\ 2 \end{pmatrix}\) but \(\displaystyle T(u+ v)=T\left(\begin{pmatrix}x_1+ x_2 \\ y_1+y_2 \\ z_1+ z_2\end{pmatrix} \right)\)\(\displaystyle = \begin{pmatrix} z_1+ z_2 \\ 1\end{pmatrix}\). That is, T(u+ v) is NOT equal to Tu+ Tv.

 

[Ereisorhet]

I did b) and it should be linear transformation if i'm correct

 

[pka]

Yes, you correctly did part b).

 

[Ereisorhet]

Thank you so much both of you @pka and @HallsofIvy