<s>[INST] Is $ T: \mathbb{R^3} -> \mathbb{R^2} : T(x,y,z) = (x+1, y+z)$ a linear transformation? Need help for my proof So I have
$$ T: \mathbb{R^3} \rightarrow\mathbb{R^2} $$
$$ T(x,y,z) = (x+1, y+z),$$
and I know that it is indeed a linear transformation if I show both cases:
$$  \forall v \in T :  F(a+b) =  F(a)+F(b),$$
and
$$  \forall k \in K: F(k \cdot a) = k \cdot F(a).$$
So I just start and I would like to know if what I'm doing is correct, I as am a little bit unsure if I'm not doing any mistakes:
$$ T(a+b)= \begin{pmatrix}  
x_1+1+x_2+1\\  
(y_1+z_1)+(y_2+z_2)  
\end{pmatrix}$$
$$ T(a)= \begin{pmatrix}  
x_1+1\\  
(y_1+z_1)  
\end{pmatrix}$$
$$ T(b)= \begin{pmatrix}  
x_2+1\\  
(y_2+z_2)  
\end{pmatrix}$$
For $T(a) + T(b)$ follows:
$$ \begin{pmatrix}  
x_1+1 + x_2+1\\  
(y_1+z_1)+(y_2+z_2)  
\end{pmatrix} = T(a+b)$$
Now for the second case, the skalar: Let $k = 2$
$$ F(k \cdot a) = \begin{pmatrix}  
2 \cdot (x_1 + 1)\\  
2 \cdot(y_1+z_1)  
\end{pmatrix} $$
now finally:
$$ k \cdot F( a) = 2 \cdot  \begin{pmatrix}  
 x_1+1\\  
(y_1+z_1)  
\end{pmatrix} =\begin{pmatrix}  
 x_1+1\\  
(y_1+z_1)  
\end{pmatrix} + \begin{pmatrix}  
 x_1+1\\  
(y_1+z_1)  
\end{pmatrix} = F(k \cdot a) $$
therefore
$$ T(x,y,z) = T(x+1, y+z),$$
is indeed a linear transformation.
I am correct or did I do something wrong? I am not sure about the skalar multiplication part but would appreciate it a lot if somebody could guide me if I'm on the right path.
Thanks in advance!
Greetings!
 [/INST] Any linear transformation can be written in the form $$T(\overline{v})=M \overline{v}$$ where $M$ is a matrix of transformation. The transformation cannot be written in this form because
$$T(x,y,z)= \begin{pmatrix} x+1 \\ x+z
\end{pmatrix}  =\begin{pmatrix} 1 & 0 & 0 \\
1 & 0 & 1
\end{pmatrix}  \begin{pmatrix} x \\ y \\ z
\end{pmatrix} + \begin{pmatrix} 1 \\ 0
\end{pmatrix} =M\overline{v}+\overline{c}$$
Hence, it is not a linear transformation.
</s>